Hopf fibrations and totally geodesic submanifolds

TL;DR

This paper classifies totally geodesic submanifolds in Hopf-Berger spheres, revealing new examples including real projective spaces and non-homogeneous submanifolds, primarily in positively curved cases.

Contribution

It provides a classification of totally geodesic submanifolds in Hopf-Berger spheres and uncovers novel examples with unique geometric properties.

Findings

Identification of totally geodesic submanifolds isometric to real projective spaces

Existence of uncountably many non-congruent totally geodesic submanifolds

Discovery of a totally geodesic submanifold that is not extrinsically homogeneous

Abstract

We classify totally geodesic submanifolds in Hopf-Berger spheres, which constitute a special family of homogeneous spaces diffeomorphic to spheres constructed via Hopf fibrations. As a byproduct of our investigations, we have discovered very intriguing examples of totally geodesic submanifolds. In particular, we stand out the following three: totally geodesic submanifolds isometric to real projective spaces, uncountably many isometric but non-congruent totally geodesic submanifolds, and a totally geodesic submanifold that is not extrinsically homogeneous. Remarkably, all these examples only arise in certain Hopf-Berger spheres with positive curvature.

Tables

Table 1. Table 1 . Well-positioned totally geodesic submanifolds of dimension d ≥ 2 𝑑 2 d\geq 2 in Hopf-Berger spheres.

${𝖲}_{ℂ,\tau }^{2n+1}$
	${𝖲}_{ℂ,\tau }^{2k+1}$	$1\le k\le n-1$
	${𝖲}_1^k$	$2\le k\le n$.aaa
${𝖲}_{ℍ,\tau }^{4n+3}$
	${𝖲}_{ℍ,\tau }^{4k+3}$	$1\le k\le n-1$
	${𝖲}_{ℂ,\tau }^{2k+1}$	$1\le k\le n$.aaa
	${𝖲}_1^k$	$2\le k\le n$.aaa
	${𝖲}_{1/\tau }^k$	$2\le k\le 3$.aaa
${𝖲}_{𝕆,\tau }^{15}$
	${𝖲}_{ℍ,\tau }^7$
	${𝖲}_{ℂ,\tau }^3$
	${𝖲}_{1/\tau }^k$	$2\le k\le 7$ aaa

Table 2. Table 2 . Not well-positioned totally geodesic submanifolds of dimension d ≥ 2 𝑑 2 d\geq 2 in  𝖲 ℍ , τ 4 ​ n + 3 subscript superscript 𝖲 4 𝑛 3 ℍ 𝜏 \mathsf{S}^{4n+3}_{\mathbb{H},\tau} for n ≥ 1 𝑛 1 n\geq 1 .

${𝖲}_{ℍ,\tau }^{4n+3}$,
	${𝖲}_{4-4\tau }^k$	$k\in \{2,3\}$
	$ℝ{𝖯}_{2/3}^2$	$\tau =1/3$
	$ℝ{𝖯}_{1-\tau }^2$	, $n\ge 2$
	$ℝ{𝖯}_{3/4}^3$	$\tau =1/4$, $n\ge 2$
	$ℝ{𝖯}_{1-\tau }^3$	, $n\ge 3$

Table 3. Table 3 . Not well-positioned totally geodesic submanifolds of dimension d ≥ 2 𝑑 2 d\geq 2 in  𝖲 𝕆 , τ 15 subscript superscript 𝖲 15 𝕆 𝜏 \mathsf{S}^{15}_{\mathbb{O},\tau} .

${𝖲}_{𝕆,\tau }^{15}$,
	${𝖲}_{4-4\tau }^k$	$k\in \{2,5,6,7\}$
	${𝖲}_{4-4\tau }^3$	Infinitely many congruence classes
	${𝖲}_{4-4\tau }^4$	Not extrinsically homogeneous
	$ℝ{𝖯}_{2/3}^2$	$\tau =1/3$

Table 4. Table 4 . Totally geodesic submanifolds in symmetric spaces of rank one and non-constant sectional curvature, up to congruence.

$ℂ{𝖯}^n$
	$ℂ{𝖯}^k$	$2\le k\le n-1$
	$ℝ{𝖯}^k$	$2\le k\le n$……
	${𝖲}_4^k$	$1\le k\le 2$…
$ℍ{𝖯}^n$
	$ℍ{𝖯}^k$	$2\le k\le n-1$
	$ℂ{𝖯}^k$	$2\le k\le n$……
	$ℝ{𝖯}^k$	$2\le k\le n$……
	${𝖲}_4^k$	$1\le k\le 4$……
$𝕆{𝖯}^2$
	$ℍ{𝖯}^2$
	$ℂ{𝖯}^2$
	$ℝ{𝖯}^2$
	${𝖲}_4^k$	$1\le k\le 8$

Table 5. Table 5 . Eigenvalues and multiplicities of the Jacobi operator of 𝖲 𝔽 , τ n subscript superscript 𝖲 𝑛 𝔽 𝜏 \mathsf{S}^{n}_{\mathbb{F},\tau} associated with unit vectors X 𝑋 X and Y 𝑌 Y in 𝔭 1 subscript 𝔭 1 \mathfrak{p}_{1} and 𝔭 2 subscript 𝔭 2 \mathfrak{p}_{2} , respectively.

	$R_X$	$R_Y$
${𝔭}_1$	$(0,1)$	$(\tau ,dim\left(𝔽\right)-1)$
	$(1/\tau ,dim\left(𝔽\right)-2)$
${𝔭}_2$	$(\tau ,n-dim\left(𝔽\right)+1)$	$(0,1)$
		$(4-3\tau ,dim\left(𝔽\right)-1)$
		$(1,n-2dim\left(𝔽\right)+1)$

Equations

$${\mathsf{S}}^1\to {\mathsf{S}}^{2n+1}\to \mathbb{C}{\mathsf{P}}^n,{\mathsf{S}}^3\to {\mathsf{S}}^{4n+3}\to \mathbb{H}{\mathsf{P}}^n,{\mathsf{S}}^7\to {\mathsf{S}}^{15}\to \mathbb{O}{\mathsf{P}}^1.$$

$${\frac{D}{\partial s}}_{|s=0}{\dot{\gamma }}_{v(s)}(t)=-{\mathcal{S}}_{\xi }^{t}J(t),$$

$${\mathcal{S}}_{\xi }^{t}J(t)=-\frac{D}{\partial t}J(t).$$

$$T_q\Sigma =W_1(q)\oplus W_2(q),\text{for all q∈Σ},$$

$$2⟨U(X,Y),Z⟩=⟨[Z,X{]}_{\mathfrak{p}},Y⟩+⟨X,[Z,Y{]}_{\mathfrak{p}}⟩,\text{where X,Y,Z∈p,}$$

$$\begin{gathered} \left\{\text{parbox}[c]104.07117pt\text{centering}Normal \\ homogeneousspaces\text{@add@centering}\right\}⊊\left\{\text{parbox}[c]104.07117pt\text{centering}Naturallyreductive \\ homogeneousspaces\text{@add@centering}\right\}⊊\left\{\text{parbox}[c]73.7146pt\text{centering}Geodesicorbit \\ spaces\text{@add@centering}\right\}. \end{gathered}$$

$$({\nabla }_{X^{\ast }}^c{Y}^{\ast }{)}_o=(-[X,Y{]}_{\mathfrak{p}}{)}_o^{\ast },\text{where X,Y∈p.}$$

$$({\nabla }_{X^{\ast }}{Y}^{\ast }{)}_o={\left(-\frac{1}{2}[X,Y{]}_{\mathfrak{p}}+U(X,Y)\right)}_o^{\ast },\text{where X,Y∈p.}$$

$$D_{X}Y=\frac{1}{2}[X,Y{]}_{\mathfrak{p}}+U(X,Y),\text{for every X,Y∈p.}$$

$$R_o(X,Y)Z$$

$$-\frac{1}{2}U([Z,Y{]}_{\mathfrak{p}},X)-\frac{1}{2}[U(Z,Y),X{]}_{\mathfrak{p}}+U(U(Z,Y),X)-\frac{1}{4}[[Z,X{]}_{\mathfrak{p}},Y{]}_{\mathfrak{p}}$$

$$+\frac{1}{2}U([Z,X{]}_{\mathfrak{p}},Y)+\frac{1}{2}[U(Z,X),Y{]}_{\mathfrak{p}}-U(U(Z,X),Y),$$

$$({\nabla }_{V}R)(X,Y,Z)$$

$$=D_{V}R(X,Y)Z-R(D_{V}X,Y)Z-R(X,D_{V}Y)Z-R(X,Y)D_{V}Z,$$

$$T_{{\gamma }_v(t)}{\mathsf{S}}_t(p)={\mathbb{V}}_1^t(v)\oplus {\mathbb{V}}_2^t(v).$$

$$T{\mathsf{S}}_t(p)={\mathcal{D}}_1^t\oplus {\mathcal{D}}_2^t.$$

$$\beta_{1}^{t}=\left\{\begin{array}[]{ll}-2\cot(2t)&\mbox{if }\bar{M}\mbox{ is of compact type,}\\ -2\coth(2t)&\mbox{if }\bar{M}\mbox{ is of non-compact type.}\end{array}\right.$$

$$\beta_{2}^{t}=\left\{\begin{array}[]{ll}-\cot(t)&\mbox{if }\bar{M}\mbox{ is of compact type,}\\ -\coth(t)&\mbox{if }\bar{M}\mbox{ is of non-compact type.}\end{array}\right.$$

$$M\cap {\mathsf{S}}_t(p)={\overline{\exp }}_p(B_{\pi /2}^E(0)\cap T_{p}M)\cap {\mathsf{S}}_t(p),\text{where t∈(0,π/2).}$$

$$t=\arccos (\sqrt{\tau }),$$

$$t=\mathrm{arccosh}(\sqrt{\tau }),$$

$$\mathfrak{k}=\left(\begin{array}[]{c|c}Z&0\\ \hline\cr 0&0\end{array}\right),\quad\mathfrak{p}_{1}=\left(\begin{array}[]{c|c}0&0\\ \hline\cr 0&\mathrm{Im}(x)\end{array}\right),\quad\mathfrak{p}_{2}=\left(\begin{array}[]{c|c}0&v\\ \hline\cr-v^{*}&0\end{array}\right),$$

$$X_1$$

$$X_3$$

$$X_5$$

$$X_7$$

$$Y_2$$

$$Y_4$$

$$Y_6$$

$$Y_8$$

$$\mathrm{Isom}({\mathsf{S}}_{\mathbb{C},\tau }^{2n+1})\cong {\mathsf{U}}_{n+1}⋊{\mathbb{Z}}_2,\mathrm{Isom}({\mathsf{S}}_{\mathbb{C},\tau }^{4n+3})\cong {\mathsf{Sp}}_{n+1}{\mathsf{Sp}}_1,\mathrm{Isom}({\mathsf{S}}_{\mathbb{O},\tau }^{15})\cong {\mathsf{Spin}}_9,$$

$$\mathsf{K}\cdot (v_1+v_2)={\mathsf{S}}_1(r)\times {\mathsf{S}}_2(s).$$

$$\check{\mathsf{K}}\cdot (v_1+v_2)={\mathsf{S}}_1(r)\times {\mathsf{S}}_2(s).$$

$$\mathsf{K}\cdot (v_1+v_2)=\{v_1\}\times {\mathsf{S}}_2(s),\check{\mathsf{K}}\cdot (v_1+v_2)={\mathsf{S}}_1(r)\times {\mathsf{S}}_2(s).$$

$$D_Y{X}_i$$

$$D_{J_{i}Y}{J}_{j}Y$$

Full text

On the one hand, the first author was supported by Famaf-UNC and CIEM-Conicet. On the other hand, the second author has been supported by the projects PID2019-105138GB-C21, PID2022-138988NB-I00 (MCIN/AEI/10.13039/501100011033/FEDER, EU, Spain) and ED431F 2020/04 (Xunta de Galicia, Spain), the FPU program (Ministry of Education, Spain), the Methusalem grant METH/15/026 of the Flemish Government (Belgium), and the FWO Postdoctoral grant with project number 1262324N.

1. Introduction

During the 1930s, Heinz Hopf [23, 24] introduced the fibrations bearing his name. These are constructed using complex numbers $\mathbb{C}$, quaternions $\mathbb{H}$, and octonions $\mathbb{O}$ and comprise the following:

[TABLE]

We will refer to these fibrations as the complex, quaternionic, or octonionic Hopf fibrations, respectively. These fibrations play a fundamental role in both geometry and topology. From a topological perspective, they contributed to the area of homotopy theory by providing an example of a homotopically non-trivial map from $\mathsf{S}^{3}$ to $\mathsf{S}^{2}$, which generates $\pi_{3}(\mathsf{S}^{2})=\mathbb{Z}$. Moreover, Adams [1] proved that a fiber bundle where the fiber, total space, and base space are connected spheres must be a Hopf fibration with base equal to $\mathbb{C}\mathsf{P}^{1}=\mathsf{S}^{2}$, $\mathbb{H}\mathsf{P}^{1}=\mathsf{S}^{4}$, or $\mathbb{O}\mathsf{P}^{1}=\mathsf{S}^{8}$. From a geometric point of view, Hopf fibrations have spurred the search of Riemannian manifolds with positive curvature by providing positively-curved non-symmetric homogeneous spaces, as for example, Berger spheres discovered in [5]. Furthermore, Gromoll, Grove and Wilking [19, 52] proved that a Riemannian submersion from a round sphere with connected fibers of positive dimension is metrically congruent to a Hopf fibration.

The purpose of this article is two-fold. Firstly, to classify totally geodesic submanifolds in Hopf-Berger spheres, which constitute a special family of homogeneous spaces diffeomorphic to spheres constructed using Hopf fibrations. Secondly, to provide certain general tools for the study of totally geodesic submanifolds in homogeneous spaces.

Let us equip the base and the total spaces of the Hopf fibrations with the corresponding symmetric metrics of diameter $\pi/2$, and sectional curvature equal to $1$, respectively. Under these conditions Hopf fibrations become Riemannian submersions with totally geodesic fibers. Let us consider, for each $\tau>0$, the total space of the complex, quaternionic, or octonionic Hopf fibration endowed with the Riemannian metric obtained from rescaling the metric tensor of the round sphere by a factor $\tau$ in the vertical directions. Such Riemannian homogeneous spaces are denoted by $\mathsf{S}_{\mathbb{C},\tau}^{2n+1}$, $\mathsf{S}_{\mathbb{H},\tau}^{4n+3}$, and $\mathsf{S}_{\mathbb{O},\tau}^{15}$, depending on whether the Hopf fibration considered is the complex, the quaternionic, or the octonionic one, respectively. We refer to such spaces as the Hopf-Berger spheres. Also, Hopf-Berger spheres can be regarded as geodesic spheres of rank one symmetric spaces, see Subsection 2.4.

The problem of classifying totally geodesic submanifolds in a given Riemannian manifold is a classical topic in the field of submanifold geometry. In the setting of Riemannian symmetric spaces, we recall that this problem has been extensively studied. However, despite all the efforts toward a general classification, we only have classifications for symmetric spaces of rank one [55], symmetric spaces of rank two [13, 14, 28, 29, 30], exceptional symmetric spaces [33], and some special classes of totally geodesic submanifolds such as reflective ones [34, 35, 36], non-semisimple maximal ones [9], or products of rank one symmetric spaces [41]. The curvature tensor of a symmetric space is parallel under the Levi-Civita connection and can be expressed by means of an easy formula in terms of Lie brackets. Hence, the problem of classifying totally geodesic submanifolds turns out to be equivalent to classifying subspaces of the tangent spaces that satisfy the harmless-looking but extremely complicated condition of being a Lie triple system, see for instance [8, §11.1.1]. Additionally, totally geodesic submanifolds in symmetric spaces are extrinsically homogeneous, i.e. orbits of the isometry group of the ambient space. In the more general setting of Riemannian homogeneous spaces, the classification of totally geodesic submanifolds is a harder problem for a number of reasons. Firstly, in this setting, we need to look for subspaces of the tangent space that are not only invariant by the curvature tensor, which has a much more involved expression, see Equation (4), but also by all its covariant derivatives, see Theorem 2.1. Secondly, although complete totally geodesic submanifolds of homogeneous spaces are intrinsically homogeneous (see [31, Corollary 8.10]), they are not necessarily extrinsically homogeneous, see for instance [27]. This picture suggests the necessity of developing new tools to classify totally geodesic submanifolds in homogeneous spaces.

Moreover, the utilization of totally geodesic submanifolds has been of capital importance when studying positively-curved spaces. As a matter of fact, one example is the investigation of pinching constants for homogeneous spaces of positive curvature, see [40]. The reason for this is that every critical point for the sectional curvature of a totally geodesic submanifold is a critical point for the sectional curvature of the ambient space. Another example is Wilking’s connectedness principle [53, Theorem 1], which provides an obstruction for the existence of totally geodesic submanifolds in compact Riemannian manifolds with positive curvature. This has allowed to develop Grove’s program toward the obtention of rigidity results for positively-curved spaces under the presence of symmetry, which holds the ultimate hope that by relaxing the symmetry hypotheses new examples would be constructed. Fixed-point components of isometries are extrinsically homogeneous totally geodesic submanifolds (see [40, Lemma A.1]), and thus Wilking’s connectedness principle provides a very useful tool to obtain such rigidity results, and its application has been quite successful in the last years, see for instance [25] or [54]. However, in this article we present numerous totally geodesic submanifolds which are not fixed-point components of isometries (those which are not well-positioned, see Lemma 4.8); or even more, an example which is not even extrinsically homogeneous, see Theorem 7.6. This indicates that the existence of totally geodesic submanifolds is not completely linked to the presence of symmetry, and therefore the scope of Wilking’s connectedness principle could possibly be broadened.

Certainly, the setting of homogeneous spaces of positive curvature turns out to be an interesting context where to study totally geodesic submanifolds since every totally geodesic submanifold of dimension $d\geq 2$ is again a homogeneous space with positive curvature. Hopf-Berger spheres provide some of the easiest examples of non-symmetric homogeneous metrics with positive sectional curvature. Indeed, they have positive sectional curvature if and only if $\tau\in(0,4/3)$. This was proved by Volper and Berestovskii in [4, 49, 50], see also [48]. Interestingly, the condition $\tau\in(0,4/3)$ also characterizes the metrics with positive sectional curvature of the space $\mathbb{C}\mathsf{P}_{\tau}^{2n+1}$, defined analogously as the Hopf-Berger spheres but using the Riemannian submersion $\mathsf{S}^{2}\rightarrow\mathbb{C}\mathsf{P}^{2n+1}\rightarrow\mathbb{H}\mathsf{P}^{n}$, see [51] for details.

Let us revisit some widely known facts concerning totally geodesic submanifolds of round spheres. The round sphere $\mathsf{S}_{1}^{n}$ can be seen as a geodesic sphere of $\mathbb{R}^{n+1}$ with radius $1$. Furthermore, every complete totally geodesic submanifold of $\mathsf{S}_{1}^{n}$ is the intersection of a complete totally geodesic submanifold of $\mathbb{R}^{n+1}$ passing through the origin (a linear subspace) with $\mathsf{S}_{1}^{n}$. The following theorem generalizes this well-known geometric fact to the setting of 2-point homogeneous spaces and provides the classification of totally geodesic submanifolds in Hopf-Berger spheres with $\tau\geq 1/2$.

Theorem A**.**

Let $\mathsf{S}^{n}_{\mathbb{F},\tau}$, with $\tau\neq 1$ and $\mathbb{F}\in\{\mathbb{C},\mathbb{H},\mathbb{O}\}$, be a Hopf-Berger sphere, $\Sigma$ a submanifold of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ with dimension $d\geq 2$, and $\bar{M}$ the symmetric space of rank one where $\mathsf{S}^{n}_{\mathbb{F},\tau}$ is realized as a geodesic sphere. Then, if $\tau\geq 1/2$ or $\dim\Sigma\geq\dim\mathbb{F}$, the following statements are equivalent:

1. i)

$\Sigma$* is a complete totally geodesic submanifold of $\mathsf{S}^{n}_{\mathbb{F},\tau}$.* 2. ii)

$\Sigma$* is the intersection of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ (regarded as a geodesic sphere of $\bar{M}$) with a complete totally geodesic submanifold $M$ of $\bar{M}$, with dimension $d+1$, containing the center of the geodesic sphere $\mathsf{S}^{n}_{\mathbb{F},\tau}$.*

Every pair $(\Sigma,\mathsf{S}^{n}_{\mathbb{F},\tau})$ satisfying the above equivalent conditions is listed in Table 1. Conversely, each pair listed in Table 1 such that $\tau\geq 1/2$ or $\dim\Sigma\geq\dim\mathbb{F}$ corresponds to a unique congruence class of totally geodesic submanifolds of $\mathsf{S}^{n}_{\mathbb{F},\tau}$.

The condition $\tau\geq 1/2$ has an interesting geometric interpretation. It turns out that the closed geodesic balls with radius $t$ given by Equation (7) are geodesically convex if and only if $\tau\geq 1/2$. This follows since the principal curvatures of the shape operator of the geodesic spheres of the corresponding rank one symmetric spaces with respect to an outer normal vector field are non-positive, see Equations (5) and (6).

Remark 1.1*.*

Notice that for $\tau=1$ the sphere $\mathsf{S}^{n}_{\mathbb{F},\tau}$ is round. If $\tau\neq 1$, we have to assume that $\Sigma$ has dimension $d\geq 2$, since there always are non-closed geodesics in $\mathsf{S}^{n}_{\mathbb{F},\tau}$, see Lemma 5.3, and a non-closed geodesic in $\mathsf{S}^{n}_{\mathbb{F},\tau}$ cannot arise as the intersection with a totally geodesic submanifold of $\bar{M}$.

The totally geodesic submanifolds described in Theorem A satisfy a natural geometric property which will be fundamental for our study. We say that a totally geodesic submanifold $\Sigma$ of a Hopf-Berger sphere $\mathsf{S}^{n}_{\mathbb{F},\tau}$ is well-positioned if $T_{p}\Sigma=(T_{p}\Sigma\cap\mathcal{H}_{p})\oplus(T_{p}\Sigma\cap\mathcal{V}_{p})$ for every $p\in M$, where $\mathcal{H}$ and $\mathcal{V}$ denote the horizontal and vertical distributions of the corresponding Hopf fibration, respectively. The examples of totally geodesic submanifolds in Table 1 are all obtained as intersections of geodesic spheres of a rank one symmetric space with a complete totally geodesic submanifold passing through the center of the corresponding geodesic sphere. It turns out that this is equivalent to being well-positioned (see Theorem 4.4). Furthermore, every well-positioned totally geodesic submanifold of a Hopf-Berger sphere is extrinsically homogeneous, see Remark 4.7.

The classification given in Theorem A works for those Hopf-Berger spheres with $\tau\geq 1/2$. However, we managed to achieve the classification for each value of $\tau>0$. Below, we describe the classification of totally geodesic submanifolds in $\mathsf{S}^{n}_{\mathbb{F},\tau}$, for each $\mathbb{F}\in\{\mathbb{C},\mathbb{H},\mathbb{O}\}$.

Let us start with $\mathbb{F}=\mathbb{C}$. In this case totally geodesic submanifolds are well-positioned and the classification was already obtained by Torralbo and Urbano [46] when $\tau<1$. Here the classification is a consequence of Theorem A. Now we will discuss the case when $\mathbb{F}\in\{\mathbb{H},\mathbb{O}\}$, where not well-positioned totally geodesic submanifolds do arise. It is important to highlight that these examples appear only in certain positively-curved Hopf-Berger spheres (those with $\tau<\tfrac{1}{2}$).

In order to study not well-positioned totally geodesic submanifolds, we will introduce a pseudo-Riemannian metric $\upalpha$ on $\mathsf{S}^{n}_{\mathbb{F},\tau}$, with $\mathbb{F}\in\{\mathbb{H},\mathbb{O}\}$. This pseudo-Riemannian metric $\upalpha$ is induced by the second fundamental form of the embedding of the corresponding geodesic sphere into $\mathbb{H}\mathsf{P}^{n}$ and $\mathbb{O}\mathsf{P}^{2}$, respectively. In particular, we prove (see Theorem 5.7) that a complete submanifold $\Sigma$ of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ of dimension $d\geq 2$, with $\tau<\tfrac{1}{2}$, is a not well-positioned totally geodesic submanifold if and only if $T_{p}\Sigma$ is $\upalpha$-isotropic and curvature invariant for every $p\in\Sigma$. This allows to reduce the seemingly infinite number of conditions given by Theorem 2.1, to check that the Riemannian exponential map of a subspace is a totally geodesic submanifold, to just two conditions.

It turns out that a good amount of not well-positioned totally geodesic submanifolds in $\mathsf{S}^{n}_{\mathbb{F},\tau}$ are isometric to round spheres. The construction of these not well-positioned totally geodesic spheres can be described in purely geometric terms. Let us consider a Helgason sphere $\mathsf{S}_{4}^{\dim\mathbb{F}}$ of $\mathbb{F}\mathsf{P}^{n}$, with $\mathbb{F}\in\{\mathbb{H},\mathbb{O}\}$, i.e. a maximal totally geodesic sphere with maximal sectional curvature. The idea is to deform a totally geodesic hypersurface $\widehat{\Sigma}$ of $\mathsf{S}_{4}^{\dim\mathbb{F}}$ in such a way that it is $\upalpha$-isotropic and curvature invariant. As a matter of fact, we prove that every not well-positioned totally geodesic sphere is congruent to a totally geodesic submanifold of the deformation of $\widehat{\Sigma}$, see Theorem 5.10.

Theorem B**.**

Let $\Sigma$ be a complete submanifold of $\mathsf{S}^{4n+3}_{\mathbb{H},\tau}$, $n\geq 1$, of dimension $d\geq 2$. Then, $\Sigma$ is totally geodesic if and only if it is congruent to a well-positioned totally geodesic submanifold (see Table 1) or a not well-positioned totally geodesic submanifold (see Table 2).

Surprisingly, in $\mathsf{S}^{4n+3}_{\mathbb{H},\tau}$, we have discovered (see Table 2) not well-positioned totally geodesic submanifolds of $\mathsf{S}^{4n+3}_{\mathbb{H},\tau}$ isometric to $\mathbb{R}\mathsf{P}^{2}_{1-\tau}$ when $\tau\leq 1/3$; and to $\mathbb{R}\mathsf{P}^{3}_{1-\tau}$ when $\tau\leq 1/4$, where we denote by $\mathbb{R}\mathsf{P}^{\ell}_{\kappa}$ the real projective space of dimension $\ell$ with constant sectional curvature equal to $\kappa>0$. The explicit description of these examples is given in Lemma 6.1 and Lemma 6.2. These are, up to the knowledge of the authors, the first known examples of totally geodesic embeddings of real projective spaces in Riemannian manifolds diffeomorphic to spheres with positive curvature. Indeed, Wilking’s connectedness principle (see [53, Theorem 1]) implies that $\mathsf{S}^{7}$ cannot carry a metric of positive sectional curvature admitting a totally geodesic submanifold diffeomorphic to $\mathbb{R}\mathsf{P}^{k}$ with $k\geq 4$. Since we have exhibited an example of a totally geodesic submanifold diffeomorphic to $\mathbb{R}\mathsf{P}^{2}$ in $\mathsf{S}^{7}_{\mathbb{H},1/3}$, it is natural to raise the question of whether there will be examples of Riemannian metrics on $\mathsf{S}^{7}$ with positive sectional curvature admitting totally geodesic submanifolds diffeomorphic to $\mathbb{R}\mathsf{P}^{3}$.

Theorem C**.**

Let $\Sigma$ be a complete submanifold of $\mathsf{S}^{15}_{\mathbb{O},\tau}$ of dimension $d\geq 2$. Then, $\Sigma$ is totally geodesic if and only if it is congruent to a well-positioned totally geodesic submanifold listed in Table 1 or a not well-positioned totally geodesic submanifold listed in Table 3.

In this case a not well-positioned maximal totally geodesic sphere $\widehat{\Sigma}$ has dimension $7$. We define a $3$-form $\upvarphi$ in $\mathsf{S}^{15}_{\mathbb{O},\tau}$ that plays an important role in the study of the congruence of the not well-positioned totally geodesic $3$-dimensional spheres. The $3$-form $\upvarphi$ is such that its restriction to the tangent space of a not well-positioned totally geodesic $7$-dimensional sphere $\widehat{\Sigma}$ agrees with the associative calibration defined by Harvey and Lawson in its foundational article on calibrated geometries, see [20]. Moreover, the absolute value of $\upvarphi$ associates a real number in $[0,1]$ to the tangent space of a not well-positioned totally geodesic $3$-dimensional sphere in such a way that two not well-positioned totally geodesic $3$-dimensional spheres $\Sigma_{1}^{3}$ and $\Sigma_{2}^{3}$ passing through some point $o\in\mathsf{S}^{15}_{\mathbb{O},\tau}$ are congruent in $\mathsf{S}^{15}_{\mathbb{O},\tau}$ if and only if $|\upvarphi(T_{o}\Sigma^{3}_{1})|=|\upvarphi(T_{o}\Sigma^{3}_{2})|$, see Proposition 7.3 and Example 7.4. All not well-positioned totally geodesic $3$-dimensional spheres are mutually isometric. In particular, this implies that there exists a family of uncountable isometric and non-congruent totally geodesic submanifolds in $\mathsf{S}^{15}_{\mathbb{O},\tau}$, when $\tau<1/2$. This is the first such family in a homogeneous space described in the literature up to the knowledge of the authors.

As we pointed out, a totally geodesic submanifold of a homogeneous space must be intrinsically homogeneous but this does not imply that it is an orbit of some subgroup of the isometry group of the ambient space. Indeed, we prove that every totally geodesic submanifold of a Hopf-Berger sphere is extrinsically homogeneous except not well-positioned totally geodesic $4$-dimensional spheres in $\mathsf{S}^{15}_{\mathbb{O},\tau}$, see Remark 4.7, Corollary 6.4 and Theorem 7.6. Hopf-Berger spheres are g.o. spaces, i.e.  spaces such that all their geodesics are orbits. Thus, this example shows that the class of homogeneous spaces where all the totally geodesic submanifolds are orbits is strictly contained in the class of g.o. spaces, and as far as the authors are aware, this is the first example of a not extrinsically homogeneous totally geodesic submanifold in a geodesic orbit space described in the literature.

This article is organized as follows. In Section 2, we summarize some standard facts about totally geodesic submanifolds, geodesic spheres and Riemannian submersions (Subsection 2.1), we fix some notation concerning homogeneous spaces (Subsection 2.2), we explain some well-known facts about rank one symmetric spaces and their totally geodesic submanifolds (Subsection 2.3), and we summarize some aspects of the geometry of Hopf-Berger spheres (Subsection 2.4). The purpose of Section 3 is to provide some general tools for studying totally geodesic submanifolds in homogeneous spaces. In particular, we prove the following three lemmas: Lemma 3.1, which gives a simple sufficient condition to check that a submanifold is totally geodesic; Lemma 3.2, which provides an obstruction for the existence of totally geodesic submanifolds in compact g.o. spaces, and Lemma 3.4, which relates the number of isotropy-congruence classes to the homogeneity of a totally geodesic submanifold. In Section 4, we characterize well-positioned totally geodesic submanifolds in Hopf-Berger spheres (see Theorem 4.4) as intersections of totally geodesic submanifolds of a rank one symmetric space $\bar{M}$ with a Hopf-Berger sphere regarded as a geodesic sphere of $\bar{M}$. Moreover, in Subsection 4.1, we analyze some examples of well-positioned totally geodesic submanifolds. In Section 5, we study not well-positioned totally geodesic submanifolds. Subsection 5.1 is devoted to the $1$-dimensional examples, i.e. geodesics, and Subsection 5.2 is devoted to give a characterization (Theorem 5.7) of not well-positioned totally geodesic submanifolds of Hopf-Berger spheres by means of the pseudo-Riemannian metric $\upalpha$ and the curvature tensor. Furthermore, in Subsection 5.3, we completely characterize not well-positioned maximal totally geodesic spheres (see Theorem 5.10). In Section 6, we construct and classify totally geodesic projective spaces in $\mathsf{S}^{4n+3}_{\mathbb{H},\tau}$ (see Lemma 6.1 and Lemma 6.2), which allows us to prove Theorem B. Finally, in Section 7, we analyze the moduli space of congruence classes of not well-positioned totally geodesic spheres and the extrinsic homogeneity of these submanifolds. This will lead us to prove that there exists a family with uncountably many isometric non-congruent totally geodesic submanifolds (see Theorem 7.2), and that every not well-positioned totally geodesic sphere of dimension $4$ is not extrinsically homogeneous (see Theorem 7.6). Thus, we can conclude the proof of Theorem C.

2. Preliminaries

2.1. Totally geodesic submanifolds, geodesic spheres, and Riemannian submersions

Let $(\bar{M},\langle\cdot,\cdot\rangle)$ be a complete Riemannian manifold. We denote by $\bar{\nabla}$ its Levi-Civita connection, by $\bar{R}$ its curvature tensor defined by the convention $\bar{R}(X,Y)=[\bar{\nabla}_{X},\bar{\nabla}_{Y}]-\bar{\nabla}_{[X,Y]}$, and by $\operatorname{inj}(p)$ its injectivity radius at $p\in\bar{M}$.

Let $M$ be an immersed submanifold of $\bar{M}$. We denote its induced Levi-Civita connection by $\nabla$, and its second fundamental form by $I\!I$. A connected submanifold $M$ is totally geodesic if every geodesic of $M$ is also a geodesic of $\bar{M}$. Equivalently, a connected submanifold $M$ of $\bar{M}$ is totally geodesic if and only if $I\!I=0$. Moreover, every totally geodesic submanifold $M$ of $\bar{M}$ can be extended to a complete totally geodesic submanifold of $\bar{M}$, see Theorem 2.1. Given a complete totally geodesic submanifold $M$ of $\bar{M}$ passing through $p\in\bar{M}$, we have $\overline{\exp}_{p}(V)=M$, for some linear subspace $V$ of $T_{p}\bar{M}$, where ${\overline{\exp}}_{p}$ denotes the Riemannian exponential map of $\bar{M}$ at $p\in\bar{M}$. We say that a subspace $V$ of $T_{p}\bar{M}$ is a totally geodesic subspace if $\overline{\exp}_{p}(V)$ is a totally geodesic submanifold of $\bar{M}$.

Indeed, one has the following result, which characterizes those subspaces of the tangent space whose image under the exponential map are totally geodesic, see [8, Sections 10.3.1 and 10.3.2].

Theorem 2.1**.**

Let $\bar{M}$ be a real analytic complete Riemannian manifold, $p\in\bar{M}$ and $V$ a linear subspace of $T_{p}\bar{M}$. There exists a complete totally geodesic immersed submanifold $M$ of $\bar{M}$ such that $p\in M$ and $T_{p}M=V$ if and only if $(\bar{\nabla}^{k}\bar{R})_{p}$ leaves $V$ invariant for every $k\geq 0$.

The above well-known theorem follows from the main theorem in [22] (cf. [8, Section 10.3.2]) and the following observation that uses the analyticity of $\bar{M}$: if $V\subset T_{p}\bar{M}$ is the tangent space of a totally geodesic submanifold at $p\in\bar{M}$ and $\rho:=\operatorname{inj}(p)$, then $\Sigma={\exp}_{p}(V_{\rho})$ is totally geodesic in $\bar{M}$, where $V_{\rho}=\{v\in V:\|v\|<\rho\}$.

In what follows, we give an overview of some well-known facts about the extrinsic geometry of geodesic spheres in Riemannian manifolds. Let $v(s)$, $s\in(-\varepsilon,\varepsilon)$, be a smooth curve in the unit tangent sphere $\mathsf{S}(T_{p}\bar{M})$ and let us consider the geodesic variation $\Gamma(s,t):=\gamma_{v(s)}(t)=\overline{\exp}_{p}(tv(s))$. Let us denote by $\tfrac{D\,}{\partial t}$ and $\tfrac{D\,}{\partial s}$ the partial covariant derivatives. Then, if $0<t<{\operatorname{inj}}(p)$, the field along $\gamma_{v(s)}(t)$ given by $\dot{\gamma}_{v(s)}(t)=\tfrac{\partial\,}{\partial t}\gamma_{v(s)}(t)$ is the outer unit normal at $\gamma_{v(s)}(t)$ to the geodesic sphere $\mathsf{S}_{t}(p):=\overline{\exp}_{p}(t\mathsf{S}(T_{p}\bar{M}))$ of radius $t>0$ and center $p\in\bar{M}$. Since $\|\dot{\gamma}_{v(s)}(t)\|=1$, we have that $\tfrac{D\,}{\partial s}_{|s=0}\dot{\gamma}_{v(s)}(t)$ and $\dot{\gamma}_{v(s)}(t)$ are orthogonal. Thus,

[TABLE]

where $\mathcal{S}_{\xi}^{t}$ denotes the shape operator of the geodesic sphere $\mathsf{S}_{t}(p)$ with respect to the outer unit direction $\xi$, and $J(t)=\tfrac{\partial}{\partial s}_{|s=0}\Gamma(s,t)$ is a Jacobi field along $\gamma_{v(0)}(t)$ with initial conditions $J(0)=0$ and $\tfrac{D\,}{\partial t}_{|t=0}J(t)=\dot{v}(0)$. Now using that $\tfrac{D\,}{\partial s}_{|s=0}\dot{\gamma}_{v(s)}(t)=\tfrac{D\,}{\partial t}J(t)$, we obtain the following expression for the shape operator of the geodesic sphere

[TABLE]

Lemma 2.2**.**

Let $\bar{M}$ be a Riemannian manifold and let $\Sigma$ be an embedded totally geodesic submanifold of $\bar{M}$. Let $M$ be an embedded submanifold of $\bar{M}$ satisfying the following conditions:

1. i)

$\Sigma\cap M$* is non-empty.* 2. ii)

$T_{q}\Sigma$* contains the normal space $\nu_{q}M$ of $M$ for every $q\in\Sigma\cap M$.*

Then, any connected component of $\Sigma\cap M$ is an embedded totally geodesic submanifold of $M$ and $T_{q}(\Sigma\cap M)$ is invariant under the shape operator $\mathcal{S}_{\xi}$ of $M$, for every $\xi\in\nu_{q}M$ and $q\in\Sigma\cap M$.

Proof.

By i) and ii), $\Sigma$ and $M$ are transverse at every $q\in\Sigma\cap M$. Then, it follows that $\Sigma\cap M$ is an embedded submanifold of $\bar{M}$. Moreover, $T_{q}(\Sigma\cap M)=(T_{q}\Sigma)\cap(T_{q}M)$ and $T_{q}\Sigma=T_{q}(\Sigma\cap M)\oplus\nu_{q}M$, for every $q\in\Sigma\cap M$.

We may assume that $\dim(\Sigma\cap M)>0$. Let $X,Y$ be tangent fields of $\Sigma\cap M$ defined around $q$. Thus $(\bar{\nabla}_{X}Y)_{q}\in T_{q}\Sigma=T_{q}(\Sigma\cap M)\oplus\nu_{q}M$. Hence, the tangent projection of $(\bar{\nabla}_{X}Y)_{q}$ to $M$ belongs to $T_{q}(\Sigma\cap M)$. This shows that $\Sigma\cap M$ is a totally geodesic submanifold of $M$. Let $\xi$ be a unit normal vector field of $M$ defined around $q\in\Sigma\cap M$ and $v\in T_{q}(\Sigma\cap M)$. Since $\Sigma$ is a totally geodesic submanifold of $\bar{M}$, $(\bar{\nabla}_{v}\xi)_{q}\in T_{q}\Sigma=T_{q}(\Sigma\cap M)\oplus\nu_{q}M$. Consequently, the projection of $(\bar{\nabla}_{v}\xi)_{q}$ to $T_{q}M$ belongs to $T_{q}(\Sigma\cap M)$, and this shows that $\mathcal{S}_{\xi_{q}}T_{q}(\Sigma\cap M)\subset T_{q}(\Sigma\cap M)$. ∎

As a consequence of Lemma 2.2 and the previous discussion, we obtain the following result.

Corollary 2.3**.**

Let $\bar{M}$ be a Riemannian manifold, $p\in\bar{M}$, and $\mathsf{S}_{t}(p)$ the geodesic sphere of radius $t\in(0,\operatorname{inj}(p))$. Let $V\subset T_{p}\bar{M}$ be a vector subspace and consider the set $V_{r}:=\{v\in V:\|v\|<r\}$, where $r\in(t,\operatorname{inj}(p))$. Assume that $\Sigma=\overline{\exp}_{p}(V_{r})$ is a totally geodesic submanifold of $\bar{M}$. Then, $\mathsf{S}_{t}(p)\cap\Sigma$ is a totally geodesic submanifold of $\mathsf{S}_{t}(p)$ whose tangent space is invariant under the shape operator $\mathcal{S}_{\xi}^{t}$ of $\mathsf{S}_{t}(p)$, where $\xi$ is the normal unit vector field of $\mathsf{S}_{t}(p)$.

The following lemma shows that there is a certain class of totally geodesic submanifolds of the total space of a given Riemannian submersion that remain totally geodesic when we rescale the metric on the vertical distribution, see  [15, Lemma 3.12] for a proof.

Lemma 2.4**.**

Let $\pi\colon M\to B$ be a Riemannian submersion and denote by $\mathcal{H}$ and $\mathcal{V}$ the horizontal and vertical distributions, respectively. Consider the family of Riemannian metrics $\langle\cdot,\cdot\rangle^{\tau}$ of $M$ obtained by rescaling the metric tensor $\langle\cdot,\cdot\rangle$ of $M$ by a factor $\tau>0$ in the vertical distribution. Moreover, let $\Sigma$ be a totally geodesic submanifold of $M$, with respect to the original metric, such that

[TABLE]

where $W_{1}(q)\subset\mathcal{H}_{q}$ and $W_{2}(q)\subset\mathcal{V}_{q}$. Then, $\Sigma$ is a totally geodesic submanifold of $M$ with respect to any metric $\langle\cdot,\cdot\rangle^{\tau}$.

2.2. Homogeneous spaces

For a nice introduction to the theory of homogeneous spaces, one can consult [2], [3, Chapter 4], or [31, Chapter X]. A Riemannian manifold $M$ is homogeneous if there exists some subgroup $\mathsf{G}$ of $\mathrm{Isom}(M)$ such that $\mathsf{G}$ acts transitively on $M$. We fix a point $o\in M$, so $M$ is diffeomorphic to $\mathsf{G}/\mathsf{K}$, where $\mathsf{K}=\mathsf{G}_{o}$ is the isotropy at $o$, by the map $\Phi\colon\mathsf{G}/\mathsf{K}\rightarrow M$ defined by $g\mathsf{K}\rightarrow g(o)$. We pull back the metric of $M$ by $\Phi$ to $\mathsf{G}/\mathsf{K}$, turning $\Phi$ into an isometry. Furthermore, the metric $\langle\cdot,\cdot\rangle$ induced in $\mathsf{G}/\mathsf{K}$ is $\mathsf{G}$-invariant.

For any $X\in\mathfrak{g}$, where $\mathfrak{g}$ is the Lie algebra of $\mathsf{G}$, we can associate a Killing vector field $X^{*}$ given by $X_{p}^{*}=\frac{d}{dt}_{|t=0}(\operatorname{Exp}(tX)\cdot p)$, for every $p\in M$, where $\operatorname{Exp}$ denotes the Lie exponential map of $\mathsf{G}$. Homogeneous spaces can be characterized in terms of Killing vector fields as follows. A Riemannian manifold $M$ is homogeneous if and only if for every $p\in M$ and every $v\in T_{p}M$, there is a Killing vector field $X\in\Gamma(TM)$ such that $X_{p}=v$. Moreover, Riemannian homogeneous spaces $\mathsf{G}/\mathsf{K}$ always have a reductive decomposition. A reductive decomposition is a splitting $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$, where $\mathfrak{k}$ is the Lie algebra of $\mathsf{K}$ and $\mathfrak{p}$ is an $\operatorname{Ad}(\mathsf{K})$-invariant subspace of $\mathfrak{g}$. Thus, we have the bracket relations $[\mathfrak{k},\mathfrak{p}]\subset\mathfrak{p}$ and $[\mathfrak{k},\mathfrak{k}]\subset\mathfrak{k}$. If we consider the linearization at $o\in M$ of the isotropy action of $\mathsf{K}$ on $M$, we get the isotropy representation of the homogeneous space $M$, which is defined as $k\in\mathsf{K}\mapsto k_{*o}\in\mathsf{GL}(T_{o}M)$, where $k_{*o}$ denotes the differential of $k$ at $o\in M$. This is equivalent to the adjoint representation of $\mathsf{G}$ restricted to $\mathsf{K}$ on $\mathfrak{p}$, since $\mathfrak{p}$ and $T_{o}M$ can be identified by the map which sends $X\in\mathfrak{p}$ to $X^{*}_{o}\in T_{o}M$.

Let us denote by $X_{\mathfrak{k}}$ and $X_{\mathfrak{p}}$ the projection of $X\in\mathfrak{g}$ onto $\mathfrak{k}$ and $\mathfrak{p}$, respectively. We define a symmetric bilinear map $U\colon\mathfrak{p}\times\mathfrak{p}\rightarrow\mathfrak{p}$ by

[TABLE]

and $\langle\cdot,\cdot\rangle$ denotes the inner product on $\mathfrak{p}$ induced by the Riemannian metric on $M$. The reductive decomposition $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ is naturally reductive if $U$ is identically zero. In particular, if $U\equiv 0$, every geodesic $\gamma$ of $M$ passing through $o\in M$ is given by $t\mapsto\operatorname{Exp}(tX)$ with $X\in\mathfrak{p}$, where $\operatorname{Exp}$ denotes the Lie exponential map of $\mathsf{G}$. A homogeneous space where every geodesic is an orbit of a $1$-parameter subgroup of the isometry group is said to be a geodesic orbit space, or, for short, g. o. space. Thus, naturally reductive spaces are g. o. Moreover, we say that the reductive decomposition $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ is normal homogeneous if there exists some $\operatorname{Ad}(\mathsf{G})$-invariant inner product $q$ on $\mathfrak{g}$ such that $\langle\cdot,\cdot\rangle=q_{\rvert\mathfrak{p}\times\mathfrak{p}}$ and $\mathfrak{p}=\mathfrak{k}^{\perp}$, where $\mathfrak{k}^{\perp}$ denotes the orthogonal complement of $\mathfrak{k}$ in $\mathfrak{g}$ with respect to $q$. For further generalizations of this concept, one can consult [3, Chapter 6]. It turns out that every normal homogeneous reductive decomposition is naturally reductive. Indeed, we have the chain of strict inclusions

[TABLE]

Let us consider the canonical connection $\nabla^{c}$ associated with a reductive decomposition $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$, which is the unique $\mathsf{G}$-invariant affine connection on $M$ such that

[TABLE]

We can express the Levi-Civita connection of $M$ at $o$ as

[TABLE]

The difference tensor $D$ at $o\in M$ is defined as $D=(\nabla-\nabla^{c})_{o}$, and using the identification of $T_{o}M$ and $\mathfrak{p}$, we have

[TABLE]

Using the formula for the Levi-Civita connection and the identification of $\mathfrak{p}$ with $T_{o}M$, we can compute the curvature tensor of $M$, which is given by

[TABLE]

where $X,Y,Z\in\mathfrak{p}$. We end up this section with a remark that will be useful to compute the covariant derivatives of the curvature tensor at the base point $o\in M=\mathsf{G}/\mathsf{K}$.

Remark 2.5*.*

By [31, Proposition 2.7, Chapter X] the curvature tensor $R$ of a Riemannian homogeneous space $M=\mathsf{G}/\mathsf{K}$ satisfies $\nabla^{c}R=0$, since it is $\mathsf{G}$-invariant. Let $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be a reductive decomposition for $M=\mathsf{G}/\mathsf{K}$. Then, using the definition of the difference tensor and the identification of $\mathfrak{p}$ with $T_{o}M$ we have

[TABLE]

where $X,Y,Z,V\in\mathfrak{p}$.

2.3. Rank one symmetric spaces and their totally geodesic submanifolds

In this section we discuss some facts about totally geodesic submanifolds in symmetric spaces of rank one that will be useful later.

Let $\bar{M}=\bar{\mathsf{G}}/\bar{\mathsf{K}}$ be a connected Riemannian symmetric space, where $\bar{\mathsf{G}}$ is up to some finite quotient equal to $\mathrm{Isom}^{0}(\bar{M})$, the connected component of the identity of the isometry group of $\bar{M}$, and $\bar{\mathsf{K}}$ is the isotropy at some fixed point $p\in\bar{M}$. Let $s_{q}$ be the geodesic reflection of $\bar{M}$ at $q\in\bar{M}$. It turns out that a submanifold $\Sigma$ of $\bar{M}$ is a complete totally geodesic submanifold of $\bar{M}$ if and only if $s_{q}\Sigma=\Sigma$ for every $q\in\Sigma$. Recall that the rank of a symmetric space is defined as the dimension of a maximal flat totally geodesic submanifold.

The complete Riemannian manifolds with non-zero constant sectional curvature are: real hyperbolic spaces, round spheres and real projective spaces. These are symmetric spaces of rank one. Furthermore, the remaining symmetric spaces of rank one are hyperbolic and projective spaces over $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$. The totally geodesic submanifolds in symmetric spaces of rank one were classified by Wolf in [55], see Figure 4.

Let $\kappa\in(0,\infty)$. We denote by $\mathsf{S}^{n}_{\kappa}$, $\mathbb{R}\mathsf{P}^{n}({\kappa})$, and $\mathbb{R}\mathsf{H}^{n}({\kappa})$, the round sphere of sectional curvature $\kappa$, the real projective space of sectional curvature $\kappa$ and the real hyperbolic space with sectional curvature $-\kappa$, respectively. In the case that $\kappa=1$ or $\kappa=-1$, we write $\mathsf{S}^{n}$, $\mathbb{R}\mathsf{P}^{n}$, or $\mathbb{R}\mathsf{H}^{n}$, respectively.

Now we assume that $\bar{M}=\bar{\mathsf{G}}/\bar{\mathsf{K}}$ has rank one and that the minimal absolute value of its sectional curvature is equal to $1$. One has that $\bar{\mathsf{K}}$, via the isotropy representation, acts transitively on $\mathsf{S}(T_{p}\bar{M})$, the unit sphere of $T_{p}\bar{M}$. Moreover, $\bar{\mathsf{K}}$ acts transitively on the geodesic sphere $\mathsf{S}_{t}(p)$ of radius $t$ centered at $p\in\bar{M}$, for each $t\in(0,\operatorname{inj}(p))$. This implies that geodesic spheres are $\bar{\mathsf{K}}$-homogeneous spaces. Recall that the Jacobi operator $\bar{R}_{v}$ associated with $v\in T_{p}\bar{M}$ is defined by $\bar{R}_{v}(X)=\bar{R}(X,v,v)$ for every $X\in T_{p}\bar{M}$. In this case, the Jacobi operators $\{\bar{R}_{v}:v\in T_{p}\bar{M},\,\|v\|=t\}$ are all conjugate by elements of $\bar{\mathsf{K}}$.

Let us assume that $\bar{M}$ does not have constant sectional curvature. It is well known that, for any non-zero $v\in T_{p}\bar{M}$, the Jacobi operator $\bar{R}_{v}$ restricted to $T_{p}\bar{M}\ominus\mathbb{R}v$ has only two different eigenvalues $\lambda_{1}$ and $\lambda_{2}$, where $\ominus$ denotes the orthogonal complement. Let $v\in T_{p}\bar{M}$ be of unit length. Then, $\lambda_{1}=\pm 4,\ \lambda_{2}=\pm 1,$ where the signs are positive if $\bar{M}$ is of compact type, and negative if $\bar{M}$ is of non-compact type. Observe that $\bar{R}_{v}$ has exactly one more eigenvalue that is equal to zero and has associated eigenspace $\mathbb{R}v$. Let $\mathbb{V}_{i}(v)$ be the eigenspace of $\bar{R}_{v}$ associated with $\lambda_{i}$ for each $i\in\{1,2\}$. When $\bar{M}$ is of compact type, ${\operatorname{inj}}(\bar{M})=\pi/2$. However, if $\bar{M}$ is of non-compact type, then it follows by the Cartan-Hadamard Theorem that $\operatorname{inj}(\bar{M})=+\infty$.

Fix $p\in\bar{M}$. For each $i\in\{1,2\}$ we define $\mathbb{V}_{i}^{t}(v)$ to be the parallel transport of $\mathbb{V}_{i}(v)$ along the unit speed geodesic $\gamma_{v}\colon[0,\operatorname{inj}(\bar{M}))\rightarrow\bar{M}$ with initial conditions $p\in\bar{M}$ and $v\in T_{p}\bar{M}$. The equation $\bar{\nabla}\bar{R}=0$ implies that $\mathbb{V}_{i}^{t}(v)$ is the eigenspace of $\bar{R}_{\dot{\gamma}_{v}(t)}$ associated with $\lambda_{i}$, for each $i\in\{1,2\}$. Moreover, $\mathbb{V}_{i}^{t}(v)\perp\dot{\gamma}_{v}(t)$ and then we have the following orthogonal decomposition

[TABLE]

Let $\bar{\mathsf{K}}_{v}$ be the isotropy of $\bar{\mathsf{K}}$ at $v$. Since $\bar{\mathsf{K}}$ preserves the curvature tensor $\bar{R}$ of $\bar{M}$ at $p$, then $\bar{\mathsf{K}}_{v}$ commutes with the Jacobi operator $\bar{R}_{v}$ and therefore leaves $\mathbb{V}_{i}(v)$ invariant. Notice that $\bar{\mathsf{K}}_{v}$ coincides with the isotropy $\bar{\mathsf{K}}_{\gamma_{v}(t)}$ of the action of $\bar{\mathsf{K}}$ on the geodesic sphere $\mathsf{S}_{t}(p)$ at $\gamma_{v}(t)$ for every $t\in(0,\operatorname{inj}(\bar{M}))$. Moreover, $\bar{\mathsf{K}}_{\gamma_{v}(t)}$ leaves invariant $\mathbb{V}_{i}^{t}(v)$ since $\bar{\mathsf{K}}_{\gamma_{v}(t)}$ and $\bar{R}_{\dot{\gamma}_{v}(t)}$ commute. Hence, for any fixed $t\in(0,\operatorname{inj}(\bar{M}))$, the subspace $\mathbb{V}_{i}^{t}(v)$ extends to a $\bar{\mathsf{K}}$-invariant distribution $\mathcal{D}_{i}^{t}$ on the geodesic sphere $\mathsf{S}_{t}(p)$. Thus,

[TABLE]

The distributions $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ coincide with the vertical and horizontal distributions, respectively, defined by the Hopf fibrations.

Remark 2.6*.*

Using Equation (1) and solving the Jacobi Equation (see e.g. [8, p. 393]) explictly, one can prove that $\mathbb{V}^{t}_{1}(v)$ and $\mathbb{V}^{t}_{2}(v)$ are the eigenspaces of the shape operator $\mathcal{S}^{t}$ with respect to the outer normal unit direction of $\mathsf{S}_{t}(p)$ in $\bar{M}$ with corresponding eigenvalues $\beta^{t}_{1}$ and $\beta^{t}_{2}$, respectively. These eigenvalues are given by the following expressions (see e.g. [11, p. 16]):

[TABLE]

[TABLE]

Notice that $\beta_{1}^{t}\neq\beta_{2}^{t}$ for every $t\in(0,\operatorname{inj}(\bar{M}))$. Moreover, since the connected group $\bar{\mathsf{K}}$ acts isometrically on $\mathsf{S}_{t}(p)$, the shape operator $\mathcal{S}^{t}$ is a $\bar{\mathsf{K}}$-invariant tensor. Then, $\mathcal{D}^{t}_{i}$ is an $\mathcal{S}^{t}$-invariant distribution of $\mathsf{S}_{t}(p)$ and $\mathcal{S}^{t}_{|\mathcal{D}_{i}^{t}}=\beta_{i}^{t}\operatorname{Id}_{|\mathcal{D}_{i}^{t}}$, for each $i\in\{1,2\}$.

Remark 2.7*.*

If $\bar{M}$ is of compact type, it is well known that $\mathbb{V}_{1}(v)\oplus\mathbb{R}v\subset T_{p}\bar{M}$ is the tangent space of a totally geodesic sphere of $\bar{M}$ isometric to $\mathsf{S}_{4}^{m+1}$, which we denote by $\mathsf{S}^{m+1}(v)$, where $m=\operatorname{rank}(\mathcal{D}^{t}_{1})$. This is a so-called Helgason sphere, see [21] and [39] for more information on Helgason spheres.

This implies that $\mathcal{D}^{t}_{1}$ is an integrable distribution of $\mathsf{S}_{t}(p)$ whose integral submanifold through $\gamma_{w}(t)=\overline{\exp}(tw)$, where $w\in T_{p}\bar{M}$ is of unit length, is the intersection of $\mathsf{S}^{m+1}(w)$ with $\mathsf{S}_{t}(p)$. Moreover, by Corollary 2.3, $\mathcal{D}^{t}_{1}$ is an autoparallel distribution of $\mathsf{S}_{t}(p)$. The same is true when $\bar{M}$ is of non-compact type. But in this case $\mathbb{V}_{1}(v)\oplus\mathbb{R}v$ is the tangent space at $p$ of a totally geodesic hyperbolic space of $\bar{M}$ that is isometric to $\mathbb{R}\mathsf{H}^{m+1}(4)$.

We finish this section with a remark that will be useful in Section 4.

Remark 2.8*.*

Let $\bar{M}=\bar{\mathsf{G}}/\bar{\mathsf{K}}$ be a simply connected compact symmetric space with non-constant sectional curvature and minimal sectional curvature $1$. Recall that $\operatorname{inj}(\bar{M})$ is $\pi/2$. Let $p\in\bar{M}$ and let $M$ be a complete totally geodesic submanifold of $\bar{M}$ that contains $p$. Then, $M$ is a rank one symmetric space of compact type with $\operatorname{inj}(M)=\pi/2$ or a closed geodesic of length $\pi$.

Let $B^{E}_{\pi/2}(0)$ be the Euclidean open ball of $T_{p}\bar{M}$ of radius $\pi/2$ centered at the origin. Hence, $\overline{\exp}_{p}\colon B^{E}_{\pi/2}(0)\to B_{\pi/2}(p)$ is a diffeomorphism, where $B_{\pi/2}(p)$ is the open ball of $\bar{M}$ with center $p$ and radius $\pi/2$. Now, since $\operatorname{inj}(M)=\pi/2$, the points $x$ of $M$ that are in the complement of $\overline{\exp}_{p}(B^{E}_{\pi/2}(0)\cap T_{p}M)$ must be at a distance of $\pi/2$ from $p$ in $\bar{M}$. Indeed, if $\gamma\colon[0,\pi/2]\to M$ is a minimizing unit speed geodesic in $M$ from $p$ to $x$, we have that $\gamma$ restricted to $[0,\pi/2)$ is a minimizing geodesic of $\bar{M}$, since $M$ is totally geodesic. Thus, the distance in $\bar{M}$ from $p$ to $x$ is $\pi/2$. This implies that $M\cap B_{\pi/2}(p)=\overline{\exp}_{p}(B^{E}_{\pi/2}(0)\cap T_{p}M)$ and then

[TABLE]

2.4. The geometry of Hopf-Berger spheres

In this section we summarize some facts concerning the geometry of Hopf-Berger spheres.

It is well-known that any homogeneous metric on the even-dimensional sphere $\mathsf{S}^{2n}$ is isometric to a round metric, while homogeneous metrics on odd-dimensional spheres $\mathsf{S}^{2n+1}$ of dimension bigger than three are homothetic to a metric lying in one of the following families, see [57]:

1. (1)

a $1$-parameter family of $\mathsf{U}_{n+1}$-invariant metrics on $\mathsf{S}^{2n+1}$, 2. (2)

a $1$-parameter family of $\mathsf{Sp}_{1}\mathsf{Sp}_{n+1}$-invariant metrics on $\mathsf{S}^{4n+3}$, 3. (3)

a $1$-parameter family of $\mathsf{Spin}_{9}$-invariant metrics on $\mathsf{S}^{15}$, 4. (4)

a $3$-parameter family of $\mathsf{Sp}_{n+1}$-invariant metrics on $\mathsf{S}^{4n+3}$.

It turns out that metrics in (2) lie in (4). Moreover, metrics in (1), (2) and (3) can be obtained by rescaling the round metric of the total space of a Hopf fibration in the direction of the fibers. In what follows, we will outline the construction of Hopf-Berger spheres.

Let $\{\mathsf{S}_{\mathbb{F},\tau}^{n}\}_{\tau>0}$ denote the family of complex, quaternionic or octonionic Hopf-Berger spheres of dimension $n$, according to the value of  $\mathbb{F}\in\{\mathbb{C},\mathbb{H},\mathbb{O}\}$. These were proved to be geodesic orbit spaces in [45]. This means that every geodesic of $\mathsf{S}_{\mathbb{F},\tau}^{n}$ is the orbit of a $1$-parameter subgroup of $\operatorname{Isom}(\mathsf{S}_{\mathbb{F},\tau}^{n})$. More generally, Hopf-Berger spheres were proved to be weakly-symmetric spaces in [6, Theorem 2]. A space $M$ is said weakly symmetric if given two points $p,q\in M$ one can find an isometry $\varphi$ of $M$ mapping $p$ to $q$ and $q$ to $p$. Weakly symmetric spaces are in particular geodesic orbit spaces, see [7] or [3, Theorem 5.3.6].

Furthermore, according to [56], $\mathsf{S}_{\mathbb{F},\tau}^{n}$ admits, for each $\tau>0$, a naturally reductive decomposition when $\mathbb{F}\in\{\mathbb{C},\mathbb{H}\}$. However, $\mathsf{S}_{\mathbb{\mathbb{O}},\tau}^{15}$ does not admit a naturally reductive decomposition for most $\tau\in(0,+\infty)$, see [57].

On the one hand, when $\tau=1$, Hopf-Berger spheres are round. Through this article, we denote round spheres with constant sectional curvature equal to $\kappa>0$ by $\mathsf{S}^{n}_{\kappa}$. On the other hand, every Hopf-Berger sphere with $\tau\neq 1$ is homothetic to a geodesic sphere of a rank one symmetric space $\bar{M}$, see [57]. We consider the metrics of these symmetric spaces rescaled in such a way that the minimal absolute value of their sectional curvatures is equal to $1$. Notice that two geodesic spheres of $\bar{M}$ of the same radius are congruent. Furthermore, it can be proved that every geodesic sphere $\mathsf{S}(t)$ of radius $t$ in a symmetric space of rank one $\bar{M}$ can be obtained by performing a homothety (cf. [11]) of ratio $\alpha$ to $\mathsf{S}^{n}_{\mathbb{F},\tau}$, with $\mathbb{F}\in\{\mathbb{C},\mathbb{H},\mathbb{O}\}$ and $\tau\in(0,+\infty)$, where the values of $t,\alpha$ and $\tau$ are related by:

[TABLE]

The complex and quaternionic Hopf fibrations are invariant under the action of $\mathsf{U}_{n+1}$ and $\mathsf{Sp}_{n+1}\mathsf{Sp}_{1}$, respectively. The complex and quaternionic Hopf-Berger spheres can be expressed as $\mathsf{S}^{2n+1}_{\mathbb{C},\tau}=\mathsf{U}_{n+1}/\mathsf{U}_{n}$ and $\mathsf{S}^{4n+3}_{\mathbb{H},\tau}=\mathsf{Sp}_{n+1}/\mathsf{Sp}_{n}$. Now we will describe the reductive decomposition for $\mathsf{S}^{n}_{\mathbb{F},\tau}=\mathsf{G}/\mathsf{K}$, for $\mathbb{F}\in\{\mathbb{C},\mathbb{H}\}$, where the Lie groups $\mathsf{G}$ and $\mathsf{K}$ are as above. Let $\mathfrak{g}$ be the Lie algebra of $\mathsf{G}$ for $\mathbb{F}\in\{\mathbb{C},\mathbb{H}\}$. We denote by $\mathrm{Im}(x)$ the imaginary part of an element $x\in\mathbb{F}$ and by $A^{*}$ the conjugate transpose of a matrix $A$ with entries in $\mathbb{F}\in\{\mathbb{C},\mathbb{H}\}$. Then, we consider the following subspaces of $\mathfrak{g}$:

[TABLE]

where $x\in\mathbb{F}$, $v\in\mathbb{F}^{n}$ and $Z$ belongs to $\mathfrak{u}_{n}$ or $\mathfrak{sp}_{n}$, when $\mathbb{F}=\mathbb{C}$ or $\mathbb{F}=\mathbb{H}$, respectively. Observe that $\mathfrak{k}$ is the Lie algebra of $\mathsf{K}$ and $[\mathfrak{k},\mathfrak{p}_{1}]=0$. On the one hand, $\mathfrak{p}_{1}$ is spanned by $\{X_{l}\}^{\dim_{\mathbb{R}}\mathbb{F}-1}_{l=1}$, where each $X_{l}$ is the matrix filled with zeroes except the last entry, which is equal to one of the imaginary units $i,j$ or $k$, for $X_{1},X_{2}$ or $X_{3}$, respectively. On the other hand, $\mathfrak{p}_{2}$ is identified with $\mathbb{F}^{n}$, and thus, we can consider the set $\{Y_{j}\}^{n}_{j=1}$ of $\mathfrak{p}_{2}$ where $Y_{j}$ is identified with the $j$-th element of the canonical basis of $\mathbb{F}^{n}$ for $\mathbb{F}\in\{\mathbb{C},\mathbb{H}\}$. Furthermore, the adjoint representation of $\mathsf{G}$ restricted to $\mathsf{K}$ acts on $\mathfrak{p}_{2}$ as the standard representation of $\mathsf{U}_{n}$ or $\mathsf{Sp}_{n}$, depending on whether $\mathbb{F}$ is equal to $\mathbb{C}$ or $\mathbb{H}$.

Now let $\mathbb{F}=\mathbb{O}$. The octonionic Hopf fibration is invariant under the action of $\mathsf{Spin}_{9}$ and $\mathsf{S}^{15}_{\mathbb{O},\tau}=\mathsf{Spin}_{9}/\mathsf{Spin}_{7}$. For positive integers $i,j$, such that $i<j<9$, let $E_{ij}\in\mathfrak{spin}_{9}$ be the matrix filled with zeroes except for the entries $(i,j)$ and $(j,i)$ which are equal to $1$ and $-1$, respectively. We define the following elements in $\mathfrak{g}=\mathfrak{spin}_{9}$.

[TABLE]

Then, we consider $\mathfrak{p}_{1}$ and $\mathfrak{p}_{2}$ as the subspaces of $\mathfrak{g}$ spanned by $\{X_{i}\}_{i=1}^{7}$ and $\{Y_{i}\}_{i=1}^{8}$, respectively. We define $\mathfrak{k}$ as the orthogonal complement of $\mathfrak{p}_{1}\oplus\mathfrak{p}_{2}$ with respect to the Killing form of $\mathfrak{g}=\mathfrak{spin}_{9}$. It turns out that $\mathfrak{k}\cong\mathfrak{spin}_{7}$ is the Lie algebra of the isotropy $\mathsf{K}=\mathsf{Spin}_{7}$ for the action of $\mathsf{G}=\mathsf{Spin}_{9}$ in $\mathsf{S}^{15}_{\mathbb{O},\tau}$ (cf. [48, p. 476]). Moreover, $\mathfrak{k}$ acts on $\mathfrak{p}_{1}$ as the standard representation of $\mathfrak{so}_{7}=\mathfrak{spin}_{7}$ and on $\mathfrak{p}_{2}$ as the irreducible spin representation of dimension eight.

To sum up, for each Hopf-Berger sphere $\mathsf{S}^{n}_{\mathbb{F},\tau}$, we can check that $[\mathfrak{k},\mathfrak{p}_{1}\oplus\mathfrak{p}_{2}]\subset\mathfrak{p}_{1}\oplus\mathfrak{p}_{2}$. Thus, $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$, with $\mathfrak{p}:=\mathfrak{p}_{1}\oplus\mathfrak{p}_{2}$, defines a reductive decomposition for $\mathsf{S}^{n}_{\mathbb{F},\tau}=\mathsf{G}/\mathsf{K}$.

Notice that for each $X_{i}\in\mathfrak{p}_{1}$, the map $J\colon\mathfrak{p}_{2}\rightarrow\mathfrak{p}_{2}$ given by $J_{X_{i}}Y:=[Y,X_{i}]$ defines a complex structure in $\mathfrak{p}_{2}$. In the following, we simply write $J_{i}$ to denote $J_{X_{i}}$, for each $i\in\{1,\ldots,\dim\mathbb{F}-1\}$. Let $m:=\dim\mathfrak{p}_{2}/\dim\mathbb{F}$. We consider the unique $\mathsf{K}$-invariant inner product $\langle\cdot,\cdot\rangle$ in $\mathfrak{p}$ such that $\{X_{i}/\sqrt{\tau},Y_{j},J_{i}Y_{j}\}_{i,j}^{\dim\mathbb{F}-1,m}$ and $\{X_{i}/(2\sqrt{\tau}),Y_{j}\}_{i,j}^{\dim\mathbb{F}-1,8}$ are orthonormal bases for $\mathfrak{p}$ when $\mathbb{F}\in\{\mathbb{C},\mathbb{H}\}$ and $\mathbb{F}=\mathbb{O}$, respectively. Thus, the tangent space of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ at a base point $o$ is identified with $\mathfrak{p}$ as inner product spaces. Furthermore, $\mathfrak{p}_{1}$ and $\mathfrak{p}_{2}$ can be identified with the vertical and horizontal spaces at $o\in\mathsf{S}^{n}_{\mathbb{F},\tau}$ defined by the corresponding Hopf fibration, respectively.

It can be checked that $J_{i}$ is a skew-symmetric endomorphism of $\mathfrak{p}_{2}$ with respect to the inner product $\langle\cdot,\cdot\rangle$, hence $J_{i}$ is an orthogonal complex structure for each subindex $i\in\{1,\ldots,\dim\mathbb{F}-1\}$. On the one hand, if $\mathbb{F}=\mathbb{H}$, we have the relations $J_{j}J_{i}=J_{k}$, where $(i,j,k)$ is a cyclic permutation of $(1,2,3)$. On the other hand, if $\mathbb{F}=\mathbb{O}$, the relations among the complex structures $\{J_{i}\}_{i=1}^{7}$ are given by the labeling of the Fano plane indicated in Figure 1. The Fano plane has 7 points and 7 lines. Each line $I$ contains exactly three points, and each of these triples has a cyclic ordering shown by the arrows in such a way that if $(i,j,k)$ are cyclically ordered, we have $J_{j}J_{i\rvert\mathbb{H}_{I}Y_{1}}=J_{k\rvert\mathbb{H}_{I}Y_{1}}$, where $\mathbb{H}_{I}Y_{1}:=\operatorname{span}\{Y_{1},J_{l}Y_{1}\}^{3}_{l=1}$.

Remark 2.9*.*

The isometry group of $\mathsf{S}^{n}_{\mathbb{F},\tau}$, $\tau\neq 1$, can be identified with the isotropy of the symmetric space of rank one $\bar{M}$ where $\mathsf{S}^{n}_{\mathbb{F},\tau}$ is embedded as a geodesic sphere, see [42, §8.5]. Therefore,

[TABLE]

where $\mathsf{Sp}_{n+1}\mathsf{Sp}_{1}:=\mathsf{Sp}_{n+1}\times_{\mathbb{Z}_{2}}\mathsf{Sp}_{1}$.

Remark 2.10*.*

Let $v_{1}\in\mathfrak{p}_{1}$ and $v_{2}\in\mathfrak{p}_{2}$ be non-zero elements of lengths $r$ and $s$, respectively. Moreover, denote by $\mathsf{S}_{1}(r)$ the sphere of radius $r$ of $\mathfrak{p}_{1}$, and by $\mathsf{S}_{2}(s)$ the sphere of radius $s$ of $\mathfrak{p}_{2}$. If $\mathbb{F}=\mathbb{O}$, then $\mathsf{K}$ acts polarly on $\mathfrak{p}$ and its orbits are given by

[TABLE]

When $\mathbb{F}=\mathbb{H}$, we can extend the group $\mathsf{G}=\mathsf{Sp}_{n+1}$ acting transitively on $\mathsf{S}^{4n+3}_{\mathbb{H},\tau}$ to the Lie group $\check{\mathsf{G}}=\mathsf{Sp}_{n+1}\mathsf{Sp}_{1}$ which also acts transitively and isometrically on $\mathsf{S}^{4n+3}_{\mathbb{H},\tau}$. The isotropy of the action of $\check{\mathsf{G}}$ at $o\in\mathsf{S}^{4n+3}_{\mathbb{H},\tau}$ is $\check{\mathsf{K}}=\mathsf{Sp}_{n}\mathsf{Sp}_{1}$. Then $\check{\mathsf{K}}$ acts polarly on $\mathfrak{p}$ and its orbits are given by

[TABLE]

Now, let $\mathbb{F}=\mathbb{C}$ and consider $\check{\mathsf{K}}$, the disconnected Lie group generated by $\mathsf{K}$ and $\sigma$, where $\sigma$ is the isometry lying in the isotropy of $\mathsf{S}^{2n+1}_{\mathbb{C},\tau}$ induced by the standard conjugation in $\mathbb{C}^{n+1}$. Then, $\check{\mathsf{K}}$ and $\mathsf{K}$ act polarly in $\mathfrak{p}$ and their orbits are given by

[TABLE]

We define the slope of a non-vertical vector $v\in\mathfrak{p}$ as the quotient $\tfrac{||v_{1}||}{||v_{2}||}$, where $v_{1}$ and $v_{2}$ are the vertical and horizontal projections of $v$, respectively. By convention, we say that a vertical vector has slope $\infty$. The previous discussion implies that two vectors in $\mathfrak{p}$ with the same length lie in the same orbit of the full isotropy representation of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ if and only if they have the same slope.

2.4.1. The curvature tensor of Hopf-Berger spheres

Now we compute the curvature tensor $R$ and the difference tensor $D$ of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ when $\mathbb{F}\in\{\mathbb{C},\mathbb{H}\}$. Notice that $D$ restricted to $\mathfrak{p}_{1}$ is zero and satisfies $D_{X}Y=-D_{Y}X$ for every $X,Y\in\mathfrak{p}_{2}$. Let $Y,Z,W\in\mathfrak{p}_{2}$ spanning a totally real subspace of $\mathfrak{p}_{2}$ and let $(i,j,k)$ be a cyclic permutation of $(1,2,3)$. Then, we have

[TABLE]

By using Equation (4), we have for a cyclic permutation $(i,j,k)$ of $(1,2,3)$,

[TABLE]

[TABLE]

Let us consider $R_{X}\colon\mathfrak{p}\rightarrow\mathfrak{p}$, the Jacobi operator associated with a vector $X\in\mathfrak{p}$, which is given by $R_{X}(Y)=R(Y,X)X$.

Let $X\in\mathfrak{p}_{1}$ be of unit length. By Equations (9) and (10), we deduce that $R_{X}$ leaves $\mathfrak{p}_{1}$ and $\mathfrak{p}_{2}$ invariant. Moreover, $R_{X{|\mathfrak{p}_{2}}}=\tau\operatorname{Id}_{\mathfrak{p}_{2}}$, and the eigenvalues of $R_{X{|\mathfrak{p}_{1}}}$ are $1/\tau$ and [math], with multiplicity $\dim\mathbb{F}-2$ and $1$, respectively. Let $Y\in\mathfrak{p}_{2}$ be a unit vector. By Equations (9) and (10), we deduce that $R_{Y}$ leaves $\mathfrak{p}_{1}$ and $\mathfrak{p}_{2}$ invariant. Moreover, $R_{Y{|\mathfrak{p}_{1}}}=\tau\operatorname{Id}_{\mathfrak{p}_{1}}$, and the eigenvalues of $R_{Y{|\mathfrak{p}_{2}}}$ are $4-3\tau$, $1$ and [math] of multiplicities $\dim\mathbb{F}-1$, $n-2\dim\mathbb{F}+1$ and $1$, respectively.

Now we compute the curvature tensor $R$ and the difference tensor $D$ of $\mathsf{S}^{15}_{\mathbb{O},\tau}$. We have again that $D$ restricted to $\mathfrak{p}_{1}$ is zero and satisfies $D_{X}Y=-D_{Y}X$ for every $X,Y\in\mathfrak{p}_{2}$. Let $(i,j,k)$ be an ordered triple contained in a line of the Fano plane, see Figure 1. Then, we have

[TABLE]

Moreover, by using Equation (4), we get

[TABLE]

[TABLE]

Let $X\in\mathfrak{p}_{1}$ be of unit length. By Equations (12) and (13), we deduce that $R_{X}$ leaves $\mathfrak{p}_{1}$ and $\mathfrak{p}_{2}$ invariant. Moreover, $R_{X{|\mathfrak{p}_{2}}}=\tau\operatorname{Id}_{\mathfrak{p}_{2}}$ and the eigenvalues of $R_{X{|\mathfrak{p}_{1}}}$ are $1/\tau$ and [math], with multiplicity $6$ and $1$, respectively. Let $Y\in\mathfrak{p}_{2}$ be a unit vector. By Equations (12) and (13), we have that, $R_{Y}$ leaves $\mathfrak{p}_{1}$ and $\mathfrak{p}_{2}$ invariant. Moreover, $R_{Y{|\mathfrak{p}_{1}}}=\tau\operatorname{Id}_{\mathfrak{p}_{1}}$ and the eigenvalues of $R_{Y{|\mathfrak{p}_{2}}}$ are $4-3\tau$ and [math] with multiplicities $7$ and $1$, respectively.

In Table 5 we summarize the pairs $(\lambda,m_{\lambda})$ of eigenvalues and multiplicities for the Jacobi operator.

3. Totally geodesic submanifolds and homogeneity

Let $M=\mathsf{G}/\mathsf{K}$ be a Riemannian homogeneous space. Since every $\mathsf{G}$-invariant tensor is parallel under the canonical connection $\nabla^{c}$, we have $\nabla^{c}R=0$ and $\nabla^{c}D=0$, where $R$ and $D$ denote the curvature tensor and the difference tensor of $M=\mathsf{G}/\mathsf{K}$, respectively. Thus, by Equations (2) and (3), the covariant derivatives of the curvature tensor can be expressed in terms of $D$ and $R$ (see Remark 2.5). Notice that homogeneous spaces are real analytic, see [12, Lemma 1.1]. Hence, every subspace $\mathfrak{p}_{\Sigma}$ of $\mathfrak{p}$ invariant under $D$ and $R$ is invariant under every covariant derivative of $R$, which implies by Theorem 2.1 that $\mathfrak{p}_{\Sigma}$ is the tangent space of some totally geodesic submanifold of $M$ and $\Sigma=\exp_{o}\mathfrak{p}_{\Sigma}$ is a complete totally geodesic submanifold of $M$, where $\exp_{o}$ denotes the Riemannian exponential map of $M$ at $o$. Hence, we obtain the following result.

Lemma 3.1**.**

Let $M=\mathsf{G}/\mathsf{K}$ be a Riemannian homogeneous space with base point $o\in M$ and reductive decomposition $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$. Let $\mathfrak{p}_{\Sigma}$ be a subspace of $\mathfrak{p}$ invariant under $D$ and $R$. Then, there is a complete totally geodesic submanifold $\Sigma$ of $M$ such that $T_{o}\Sigma=\mathfrak{p}_{\Sigma}$.

We remark that Lemma 3.1 provides a sufficient condition to obtain totally geodesic submanifolds in homogeneous spaces in terms of a simple linear algebraic property. However, there are many totally geodesic submanifolds whose tangent space is not invariant under the difference tensor $D$.

Now observe that the integral lines of a Killing vector field $X$ of constant length are geodesics. In fact, by making use of the Killing equation, one has that the gradient of the real function $\|X\|^{2}$ is $\frac{1}{2}\nabla_{X}X$. Then $\nabla_{X}X=0$ and every integral line of $X$ is a geodesic.

Recall that a geodesic orbit space (g.o. in short) is a Riemannian manifold $M$ such that every geodesic is the orbit of a one parameter subgroup of $\operatorname{Isom}(M)$ or, equivalently, an integral line of a Killing vector field of $M$. Observe that a totally geodesic submanifold $\Sigma$ of $M$ is also a g.o. space. This follows from the fact that the projection to $T\Sigma$ of the restriction to $\Sigma$ of a Killing vector field of $M$ is a Killing vector field of $\Sigma$.

Lemma 3.2**.**

Let $M=\mathsf{G}/\mathsf{K}$ be a compact g.o. space and let $\Sigma$ be a totally geodesic submanifold of $M$ of dimension at least $2$. Then, either $\Sigma$ is compact or $\Sigma$ has a totally geodesic and flat submanifold $F$ of dimension at least $2$.

Proof.

Let us assume that $\Sigma$ is not compact. Then, the projection of the restriction to $\Sigma$, of a Killing vector field of $M$ to $T\Sigma$ is a bounded Killing vector field since $M$ is compact. Since there is a projected Killing vector field in any direction tangent to $\Sigma$, we conclude that $\mathfrak{b}\cdot q=T_{q}\Sigma$, where $\mathfrak{b}$ denotes the ideal, in the full isometry algebra of $\operatorname{Isom}(\Sigma)$, of bounded intrinsic Killing vector fields of $\Sigma$. Let $\mathsf{B}$ be the (transitive) Lie subgroup of isometries of $\Sigma$ associated with $\mathfrak{b}$. By making use of [37, Theorem 1.3], one has that $\mathsf{B}=\mathsf{H}\mathsf{A}$ (almost direct product) where $\mathsf{H}$, if non-trivial, is compact semisimple and $\mathsf{A}$ is contained in the center of $\operatorname{Isom}(\Sigma)$ and thus it is abelian. Observe that $\dim\mathsf{A}\geq 1$ since $\Sigma$ is not compact. We claim that $\mathsf{A}$ is closed in $\operatorname{Isom}(\Sigma)$. Let $\bar{\mathfrak{a}}$ be the Lie algebra of the closure of $\mathsf{A}$ in $\operatorname{Isom}(\Sigma)$ and $q\in\Sigma$. Then, since $[\mathfrak{b},\mathfrak{a}]=0$, we have

[TABLE]

Hence, $X^{*}$ is a Killing vector field of constant length, and thus, bounded since $\mathsf{B}$ acts transitively on $\Sigma$. Then, every orbit of $\mathsf{A}$ is a totally geodesic and flat (closed) submanifold of $\Sigma$. By [18, Lemma 5.3], see also Remark 5.4 therein, $\dim\mathsf{A}\cdot q=\dim\mathsf{A}$, for all $q\in M$. Hence, if $\dim\mathsf{A}\geq 2$, we are done.

Let us assume that $\dim\mathsf{A}=1$. Since $\mathsf{B}=\mathsf{H}\mathsf{A}$ is transitive on $\Sigma$ and $\Sigma$ is not compact, the orbits of the normal subgrop $\mathsf{H}$ of $\mathsf{B}$ induces a $\mathsf{B}$-invariant foliation $\mathfrak{F}$ of codimension one on $\Sigma$. We claim that $\mathfrak{F}$ is a totally geodesic foliation. Let us prove that $\mathsf{H}\cdot q$ is a totally geodesic submanifold of $\Sigma$ for each $q\in\Sigma$. Notice that $X\in\mathfrak{b}$ induces a Killing vector field of $\Sigma$ that projects to a vector field $\bar{X}$ in the quotient $\mathsf{H}\setminus\Sigma$ given by the orbits of the action of $\mathsf{H}$ on $\Sigma$. Then, if $X_{q}\in T_{q}(\mathsf{H}\cdot q)$, we have

[TABLE]

where $\pi\colon\Sigma\rightarrow\mathsf{H}\setminus\Sigma$ denotes the quotient map. Then the restriction of $X$ to $\mathsf{H}\cdot q$ is always tangent to $\mathsf{H}\cdot q$. Hence, $\operatorname{Exp}(tX\cdot q)\in\mathsf{H}\cdot q$ for every $t\in\mathbb{R}$. This implies that $\mathsf{H}\cdot q$ is a totally geodesic submanifold of $\Sigma$ since $\Sigma$ is g.o. with respect to the presentation $\mathsf{B}/\mathsf{B}_{q}$.

Let $\mathcal{F}$ be the autoparallel distribution associated to $\mathfrak{F}$. Observe that $\mathsf{H}$ acts polarly on $\Sigma$, since it is a cohomogeneity one action. Hence, the perpendicular $1$-dimensional distribution $\mathcal{F}^{\perp}$ is autoparallel. Then $\mathcal{F}$ and $\mathcal{F}^{\perp}$ are parallel distributions. In fact, any two orthogonally complementary autoparallel distributions must be parallel. Then $\Sigma$ locally splits and it admits a $2$-dimensional totally geodesic submanifold (e.g. a product of geodesics). ∎

Remark 3.3*.*

Let $M=\mathsf{G}/\mathsf{K}$ be a compact Riemannian homogeneous manifold and let $\Sigma$ be a totally geodesic and flat (immersed) submanifold that contains the base point $o=e\cdot\mathsf{K}$. Let us consider the isotropy action of $\mathsf{K}$ on $T_{o}M$.

Then $\mathsf{K}\cdot w$ is perpendicular to $T_{o}\Sigma$ at $w$, for all $w\in T_{o}\Sigma$. In fact, let $Y$ be in the Lie algebra $\mathfrak{k}$ of $\mathsf{K}$ and let us consider the Killing vector field $Y^{*}$ induced by $Y$. Since $M$ is compact, $Y^{*}$ is bounded. Then the projection $\bar{Y}$ to $T\Sigma$ of $Y^{*}_{|\Sigma}$ is a bounded Killing vector field of $\Sigma$ which vanishes at $o$. Then $\bar{Y}=0$ and so $Y^{*}_{|\Sigma}$ is always perpendicular to $\Sigma$. Then $\nabla_{T_{q}\Sigma}Y^{*}\perp T_{q}\Sigma$ for all $q\in\Sigma$ and in particular for $q=o$. We have that $(\mathrm{Exp}(tY))_{*o}(w)=\mathrm{e}^{t(\nabla Y^{*})_{o}}(w)$ (see formula (2.2.1) in [17]). Hence, differentiating at $t=0$, we obtain that $Y.w=\nabla_{w}Y^{*}$, where $Y.w:=\tfrac{d}{dt}_{\rvert t=0}(\mathrm{Exp}(tY))_{*o}(w)$. Consequently, $Y.T_{o}\Sigma\perp T_{o}\Sigma$ for all $Y$ in the Lie algebra of $\mathsf{K}$.

Let $M=\mathsf{G}/\mathsf{K}$ be a homogeneous Riemannian manifold and let $\Sigma$ be an embedded (connected) complete totally geodesic submanifold that contains the base point $o=e\mathsf{K}$. Let $\mathsf{G}_{\Sigma}:=\{g\in\mathsf{G}:g\Sigma=\Sigma\}$ be the Lie sugroup of $\mathsf{G}$ that leaves $\Sigma$ invariant. Observe that $\mathsf{G}^{\prime}_{\Sigma}:=\{g\in\mathsf{G}:g_{|\Sigma}=\mathrm{Id}_{|\Sigma}\}$ is a normal subgroup of $\mathsf{G}_{\Sigma}$ and $\mathsf{G}_{\Sigma}/\mathsf{G}^{\prime}_{\Sigma}$ acts effectively on $\Sigma$. In the case that $\mathsf{G}$ is compact there exists a normal subgroup $\mathsf{N}$ of $\mathsf{G}_{\Sigma}$ such that $\mathsf{G}_{\Sigma}=\mathsf{G}^{\prime}_{\Sigma}\cdot\mathsf{N}$ (almost direct product) and $\mathsf{N}$ acts almost effectively on $\Sigma$.

The totally geodesic submanifold $\Sigma$ is called extrinsically homogeneous with respect to the presentation group $\mathsf{G}$ of $M$, if $\mathsf{G}_{\Sigma}$ acts transitively on $\Sigma$. Moreover, $\Sigma$ is extrinsically homogeneous if it is extrinsically homogeneous with respect to $\mathsf{G}=\operatorname{Isom}(M)$.

We say that two complete totally geodesic submanifolds $\Sigma_{1}$ and $\Sigma_{2}$ passing through $o$ are isotropy-congruent at $o$ if there exists $k\in\mathsf{K}$ such that $k\Sigma_{1}=\Sigma_{2}$ or, equivalently, if $k_{*o}T_{o}\Sigma_{1}=T_{o}\Sigma_{2}$.

Lemma 3.4**.**

Let $M=\mathsf{G}/\mathsf{K}$ be a homogeneous Riemannian manifold and $\Sigma$ be a complete totally geodesic submanifold passing through $o$. If there exist finitely many isotropy-congruence classes at $o$ of totally geodesic submanifolds isometric to $\Sigma$, then every complete embedded totally geodesic submanifold of $M$ isometric to $\Sigma$ is extrinsically homogeneous with respect to $\mathsf{G}$.

Proof.

Let $\Sigma$ be a totally geodesic submanifold of $M$ passing through $o$. Assume that the closed subgroup $\mathsf{G}_{\Sigma}:=\{g\in\mathsf{G}:g\Sigma=\Sigma\}$ does not act transitively on $\Sigma$, and let us prove that there are infinitely many isotropy-congruence classes at $o$ of totally geodesic submanifolds isometric to $\Sigma$. Then $\mathsf{G}_{\Sigma}\cdot p$ is a closed subset of $\Sigma$ for every $p\in\Sigma$. Moreover, the sets $\mathsf{G}_{\Sigma}\cdot p$ do coincide or have empty intersection. By Sard’s Theorem, each orbit of $\mathsf{G}_{\Sigma}$ on $\Sigma$ has measure zero in $\Sigma$. Since a countable union of measure zero subsets of $\Sigma$ has measure zero in $\Sigma$, there is a sequence $\{q_{n}\}_{n}\in\mathbb{N}$ in $\Sigma$ such that $q_{i}$ do not belong to $\mathsf{G}_{\Sigma}\cdot q_{j}$ for each $i\neq j$. Now take $g_{n}\in\mathsf{G}$ in such a way that $g_{n}q_{n}=o$, $n\in\mathbb{N}$, and define $\Sigma_{n}:=g_{n}\Sigma$. Let $i$ and $j$ be distinct elements in $\mathbb{N}$ and $k\in\mathsf{K}$ such that $k\Sigma_{i}=\Sigma_{j}$. Then $g^{\prime}:=g_{j}^{-1}kg_{i}\in\mathsf{G}_{\Sigma}$ and $g^{\prime}q_{i}=q_{j}$. This yields a contradiction showing that $\Sigma_{i}$ is not congruent to $\Sigma_{j}$ for every $i\neq j$, since $q_{i}$ and $q_{j}$ do not belong to the same $\mathsf{G}_{\Sigma}$-orbit. Therefore there are infinitely many congruence classes at $o$ of totally geodesic submanifolds of $M$ isometric to $\Sigma$. ∎

Remark 3.5*.*

Notice that two totally geodesic submanifolds $\Sigma_{1}$ and $\Sigma_{2}$ of $M$ passing through $o\in M$ can be congruent but not isotropy-congruent at $o\in M$. Indeed, we will see that there are not well-positioned totally geodesic $4$-dimensional spheres, which are congruent in $\mathsf{S}^{15}_{\mathbb{O},\tau}$ but not isotropy-congruent at $o\in\mathsf{S}^{15}_{\mathbb{O},\tau}$, since they are not extrinsically homogeneous, see Theorem 7.6.

4. Well positioned totally geodesic submanifolds of Hopf-Berger spheres

The goal of this section is to give a characterization of well-positioned totally geodesic submanifolds in Hopf-Berger spheres.

Motivated by Lemma 2.4, we say that a totally geodesic submanifold $\Sigma$ of a Hopf-Berger sphere $\mathsf{S}^{n}_{\mathbb{F},\tau}$ is well-positioned at $p\in\Sigma$ if $T_{p}\Sigma=(T_{p}\Sigma\cap\mathcal{H}_{p})\oplus(T_{p}\Sigma\cap\mathcal{V}_{p})$, where $\mathcal{H}$ and $\mathcal{V}$ denote the horizontal and vertical distributions of the corresponding Hopf fibration, respectively. Moreover, we say that the totally geodesic submanifold $\Sigma$ is well-positioned if the previous property holds for every $p\in\Sigma$.

The following lemma implies that if a totally geodesic submanifold has a vertical or horizontal tangent vector, then it is well-positioned at the corresponding point.

Lemma 4.1**.**

Let $\Sigma$ be a totally geodesic submanifold of $\mathsf{S}^{n}_{\mathbb{F},\tau}$, with $\tau\neq 1$. Let $\mathfrak{p}_{\Sigma}\subset\mathfrak{p}$ be identified with the tangent space of $\Sigma$ at $o\in\mathsf{S}^{n}_{\mathbb{F},\tau}$. Then, if $\mathfrak{p}_{\Sigma}\cap\mathfrak{p}_{i}\neq 0$ for some $i\in\{1,2\}$, we have $\mathfrak{p}_{\Sigma}=(\mathfrak{p}_{\Sigma}\cap\mathfrak{p}_{1})\oplus(\mathfrak{p}_{\Sigma}\cap\mathfrak{p}_{2})$ and $\Sigma$ is well-positioned at $o$.

Proof.

Let us identify $\mathfrak{p}_{\Sigma}\subset\mathfrak{p}$ with the tangent space at $o\in\mathsf{S}^{n}_{\mathbb{F},\tau}$ of a totally geodesic submanifold $\Sigma$ in $\mathsf{S}^{n}_{\mathbb{F},\tau}$. Also, if $X\in\mathfrak{p}_{\Sigma}\cap\mathfrak{p}_{1}$ is a non-zero vector, we have $R_{X}\mathfrak{p}_{\Sigma}\subset\mathfrak{p}_{\Sigma}$. Then, the sets of eigenvalues of $R_{X\rvert\mathfrak{p}_{1}}$ and $R_{X\rvert\mathfrak{p}_{2}}$ have non-trivial intersection if and only if $\tau=1$, see Table 5. If $Y\in\mathfrak{p}_{\Sigma}\cap\mathfrak{p}_{2}$ is a non-zero vector, we have $R_{Y}\mathfrak{p}_{\Sigma}\subset\mathfrak{p}_{\Sigma}$. Then, the sets of eigenvalues of $R_{Y\rvert\mathfrak{p}_{1}}$ and $R_{Y\rvert\mathfrak{p}_{2}}$ have non-trivial intersection if and only if $\tau=1$, see Table 5. Hence, if there is some non-zero vector $Z\in\mathfrak{p}_{\Sigma}\cap\mathfrak{p}_{i}$, for some $i\in\{1,2\}$, we have $\mathfrak{p}_{\Sigma}=(\mathfrak{p}_{\Sigma}\cap\mathfrak{p}_{1})\oplus(\mathfrak{p}_{\Sigma}\cap\mathfrak{p}_{2})$ when $\tau\neq 1$. ∎

Lemma 4.2**.**

Let $\Sigma$ be a complete totally geodesic submanifold of $\mathsf{S}^{n}_{\mathbb{F},\tau}$, with $\tau\neq 1$. If there exists $p\in\Sigma$ such that $\Sigma$ is well-positioned at $p$, then $\Sigma$ is well-positioned.

Proof.

For the sake of notational simplicity and since the following arguments are local, we will assume that $\Sigma$ is embedded.

Firstly, let us suppose that $T_{p}\Sigma\cap\mathcal{V}_{p}=\{0\}$. Let $q\in\Sigma$ be arbitrary and let $\gamma$ be a geodesic of $\Sigma$ with $\gamma(0)=p$, $\gamma(1)=q$. Observe that $\gamma$ is a horizontal geodesic since Hopf fibrations are Riemannian submersions and $\gamma^{\prime}(0)\in\mathcal{H}_{p}$. Hence, $\gamma^{\prime}(1)\in\mathcal{H}_{q}$. Then, by Lemma 4.1, $T_{q}\Sigma=(T_{q}\Sigma\cap\mathcal{H}_{q})\oplus(T_{q}\Sigma\cap\mathcal{V}_{q})$, and thus, $\Sigma$ is well-positioned at $q$.

Now let us assume that $T_{p}\Sigma\cap\mathcal{V}_{p}\neq\{0\}$. If $T_{p}\Sigma\subset\mathcal{V}_{p}$, then $\Sigma$ is locally contained around $p$ in the fiber $F(p)$, which is the totally geodesic integral submanifold of $\mathcal{V}$ by $p$. Then $\Sigma$ is well-positioned around $p$. Thus, we may assume that $T_{p}\Sigma\cap\mathcal{H}_{p}\neq\{0\}$. Let $\Sigma^{\prime}$ be the totally geodesic submanifold obtained by intersecting $\Sigma$ and $F(p)$ around $p$. Let $\nu(\Sigma^{\prime})$ be the normal bundle of $\Sigma^{\prime}$ as a submanifold of $\Sigma$. Observe that $\Sigma$ is well-positioned at $q$ for every element $q\in\Sigma^{\prime}$. Then $\nu_{q}(\Sigma^{\prime})$ is contained in $\mathcal{H}_{q}$. One has that there exists $\delta>0$ such that the normal exponential map from $\nu_{\delta}(\Sigma^{\prime})$ into $\Sigma$ is a diffeomorhism into its image $U\subset\Sigma$, where $\nu_{\delta}(\Sigma^{\prime})$ are the normal vectors with length less that $\delta$. This implies that any element $x\in U\setminus\Sigma^{\prime}$ is of the form $x=\gamma(1)$, with $\gamma$ a geodesic of $\Sigma$ with initial velocity in $\nu_{q}$ for some $q$. Moreover, $\gamma^{\prime}(1)\in T_{x}\Sigma\cap\mathcal{H}_{x}$. Then, by Lemma 4.1, $\Sigma$ is well-positioned at $x$. If $x\in\Sigma^{\prime}$, then $T_{x}\Sigma$ has a non-trivial vertical vector and hence, by Lemma 4.1, $\Sigma$ is well-positioned at $x$. We have proved that the points where $\Sigma$ is well-positioned form an open subset of $\Sigma$. Moreover, by continuity and Lemma 4.1, we have that the set of points of $\Sigma$ where $\Sigma$ is not well-positioned defines an open subset of $\Sigma$. Since $\Sigma$ is connected, we conclude that $\Sigma$ is well-positioned at every point, and thus, $\Sigma$ is well-positioned. ∎

Remark 4.3*.*

Let $\bar{M}$ be a symmetric space of rank one with minimal absolute value of its sectional curvature equal to one, and consider for each $t\in(0,\operatorname{inj}(p))$ the natural map $h^{t}\colon\mathsf{S}(T_{p}\bar{M})\to\mathsf{S}_{t}(p)$ given by $h^{t}(q):=\overline{\exp}_{p}(tq)=\gamma_{q}(t)$, where $\mathsf{S}_{t}(p)$ is endowed with the Riemannian metric induced by $\bar{M}$. We compute the pull-back metric $\langle\cdot,\cdot\rangle^{t}$ induced by $h^{t}$ on $\mathsf{S}(T_{p}\bar{M})$ at a point $q\in\mathsf{S}(T_{p}\bar{M})$.

Let $w\in T_{q}\mathsf{S}(T_{p}\bar{M})$ and let $c(s)$ be a curve in $\mathsf{S}(T_{p}\bar{M})$ with initial conditions $c(0)=q$ and $\dot{c}(0)=w$. Then $h^{t}(c(s))=\gamma_{c(s)}(t)$ is a variation of radial geodesics starting at $p$. Then

[TABLE]

where $J_{w}(t)$ is the Jacobi field along $\gamma_{q}(t)$ with initial conditions $J(0)=0$ and $J^{\prime}_{w}(0)=w$. Hence,

[TABLE]

There are two complementary distributions $\mathcal{F}_{1}$ and $\mathcal{F}_{2}=\mathcal{F}_{1}^{\perp}$ (with respect to $\langle\cdot,\cdot\rangle^{1}$) on $\mathsf{S}(T_{p}\bar{M})$ defined by the eigenspaces associated with the eigenvalues $\lambda_{1}=\pm 4$ and $\lambda_{2}=\pm 1$ of the Jacobi operator $\bar{R}_{\cdot,q}q$ of $\bar{M}$ at $p$ (the plus sign is for the compact case and the minus sign for the non-compact case). One has that $(h^{t})_{*}(\mathcal{F}_{i})=\mathcal{D}^{t}_{i}$, for each $i\in\{1,2\}$ (see Section 2.3 for the definition of $\mathcal{D}^{t}_{i}$). Let $\widetilde{w}_{i}$ be the parallel vector field along $\gamma_{q}$ with initial condition $\widetilde{w}_{i}(0)=w_{i}$. If $w_{i}\in\mathcal{F}_{i}(q)$, then $J_{w_{i}}(t)=\frac{1}{\sqrt{\lambda_{i}}}\sin(\sqrt{\lambda_{i}}\,t)\tilde{w}_{i}(t)$ if $\bar{M}$ is of compact type, or $J_{w_{i}}(t)=\frac{1}{\sqrt{-\lambda_{i}}}\sinh(\sqrt{-\lambda_{i}}\,t)\tilde{w}_{i}(t)$ if $\bar{M}$ is of non-compact type. In addition to that, $\langle\mathcal{F}_{1},\mathcal{F}_{2}\rangle^{t}=0$ and

[TABLE]

Recall that $\mathsf{S}^{n}_{\mathbb{F},1}$ is a round sphere of constant sectional curvature equal to $1$. Moreover, $\mathsf{S}^{n}_{\mathbb{F},\tau}$ is homothetic to the geodesic sphere $\mathsf{S}_{t}(p)$ via a homothety of ratio $\alpha=\sin(t)$ or $\alpha=\sinh(t)$, depending on whether $\bar{M}$ is of compact or non-compact type, respectively. Notice that this proves that the radius $t$ of the geodesic sphere and $\tau$ are related by $t=\arccos(\sqrt{\tau})$ in the compact setting, as we pointed out in Equation (7), since $\tau=\tfrac{\sin^{2}(2t)}{4\sin^{2}(t)}=\cos^{2}(t)$. The relation $t=\operatorname{arccosh}(\sqrt{\tau})$ in the non-compact setting is obtained analogously.

Theorem 4.4**.**

Let $\Sigma$ be a totally geodesic submanifold of $\mathsf{S}^{n}_{\mathbb{F},\tau}$, $\tau\neq 1$, and let $\bar{M}$ be the corresponding rank one symmetric space such that $\mathsf{S}^{n}_{\mathbb{F},\tau}$ arises as a geodesic sphere of $\bar{M}$ centered at $p\in\bar{M}$. Then, the following statements are equivalent:

1. i)

$\Sigma$* is a well-positioned totally geodesic submanifold of $\mathsf{S}^{n}_{\mathbb{F},\tau}$.* 2. ii)

$T_{q}\Sigma$* has a vertical or a horizontal non-zero vector for every $q\in\Sigma$.* 3. iii)

$T_{q}\Sigma$* is invariant under the shape operator of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ in $\bar{M}$ for every $q\in\Sigma$.* 4. iv)

$\Sigma$* is the intersection of a totally geodesic submanifold $M$ of $\bar{M}$ containing $p\in\bar{M}$ with $\mathsf{S}^{n}_{\mathbb{F},\tau}$ regarded as a geodesic sphere of $\bar{M}$.*

Proof.

First of all, let us identify $\mathsf{S}^{n}_{\mathbb{F},\tau}$ with the geodesic sphere $\mathsf{S}_{t}(p)$ of the appropiate symmetric space $\bar{M}$ of rank one centered at $p\in\bar{M}$ and $t=\arccos(\sqrt{\tau})$, in the compact case; or $t=\operatorname{arccosh}(\sqrt{\tau})$, in the non-compact case, see Equation (7).

Now, observe that i) and ii) are equivalent by Lemma 4.1. Moreover, let us recall that i) and iii) are equivalent since the only eigenvalue of the shape operator $\mathcal{S}^{t}$ of $\mathsf{S}_{t}(p)$ restricted to the horizontal distribution is different to the only eigenvalue of the shape operator restricted to the vertical distribution, for every $t\in(0,\operatorname{inj}(p))$, see Equations (5), (6) and the paragraph below them.

Now, by Corollary 2.3 and Remark 2.8, the intersection of a totally geodesic submanifold of $\bar{M}$ that contains $p$ with $\mathsf{S}_{t}(p)$ is totally geodesic in $\mathsf{S}_{t}(p)$ and its tangent space is invariant under $\mathcal{S}^{t}$. Hence, iv) implies iii).

Let us prove that iii) implies iv). Let $0<s,t<\operatorname{inj}(p)$ and consider the map $f_{s}:\mathsf{S}_{t}(p)\to\mathsf{S}_{s}(p)$ given by $f_{s}(\overline{\exp}_{p}(tv))=\overline{\exp}_{p}(sv)$, for all $v\in T_{p}M$ of unit length. Then the pull-back by $f_{s}$ of the Riemannian metric of $\mathsf{S}_{s}(p)$, if $M$ is of compact type, is given by modifying the metric on $\mathsf{S}_{t}(p)$ by the factor $\frac{\sin^{2}(2s)}{\sin^{2}(2t)}$ on the distribution $\mathcal{D}_{1}^{t}$ and by the factor $\tfrac{\sin^{2}(s)}{\sin^{2}(t)}$ on the distribution $\mathcal{D}_{2}^{t}$ (in the non-compact case the trigonometric functions have to be replaced by the corresponding hyperbolic functions). By Remark 4.3, if we rescale this pull-back metric by a factor of $\frac{\sin^{2}(t)}{\sin^{2}(s)}$, we are under the assumptions of Lemma 2.4. This shows that $\Sigma$ is also totally geodesic with respect to the pull-back metric for every $s\in(0,\operatorname{inj}(p))$. Or equivalently, $f_{s}(\Sigma)$ is a totally geodesic submanifold of $\mathsf{S}_{s}(p)$. Furthermore, using that $T_{q}\Sigma$ is invariant under the shape operator for every $q$ and that the integral curves of the outer unit normal vector field to the geodesic spheres are geodesics, it is standard to show that $\widehat{\Sigma}:=\bigcup_{0<s<\operatorname{inj}(p)}f_{s}(\Sigma)$ is a totally geodesic submanifold of the open ball $B_{\operatorname{inj}(p)}(p)$ which contains a piece of the radial geodesic $\gamma_{v}(t)=\overline{\exp}_{p}(tv)$. Since a totally geodesic submanifold of a symmetric space extends to a complete totally geodesic submanifold, the complete extension $\widetilde{\Sigma}$ of $\widehat{\Sigma}$ contains $p$. By making use of Remark 2.8, we have $\Sigma=\widetilde{\Sigma}\cap\mathsf{S}_{t}(p)$, and it follows that iii) implies iv). ∎

Notice that if $\dim(\Sigma)\geq\dim\mathbb{F}$, then $T_{p}\Sigma$ has a horizontal non-zero vector for every $p\in\Sigma$. Thus, by combining Lemma 4.1 and Theorem 4.4 we have the following:

Corollary 4.5**.**

Let $\Sigma$ be a totally geodesic submanifold of $\mathsf{S}^{n}_{\mathbb{F},\tau}$. If $\dim\Sigma\geq\dim\mathbb{F}$, then $\Sigma$ is well-positioned.

Remark 4.6*.*

Firstly, observe that the intersection between a geodesic sphere and a totally geodesic submanifold passing through the origin is transverse. Hence, the well-positioned totally geodesic submanifolds of Hopf-Berger spheres are embedded.

Secondly, notice that the intersection of a complete totally geodesic submanifold $M$ of $\bar{M}$ passing through $p$ with a geodesic sphere $\mathsf{S}_{t}(p)$ of $\bar{M}$ is a geodesic sphere of ${M}$ of radius $t>0$. Indeed, let us denote by $\mathsf{S}^{\prime}_{t}(p)$ the geodesic sphere of $M$ of radius $t>0$ centered at $p$. Let us denote by $d_{M}$ and $d_{\bar{M}}$ the distances on $M$ and $\bar{M}$ induced by the respective Riemannian metrics. Then,

[TABLE]

where we have used that $M$ is a complete totally geodesic submanifold of $\bar{M}$.

Remark 4.7*.*

Notice that the isotropy of $\bar{M}$ at $p$ is equal to $\operatorname{Isom}(\mathsf{S}^{n}_{\mathbb{F},\tau})$, see Remark 2.9. Moreover, for each $i\in\{1,2\}$, let $\mathsf{S}_{i}(t)$ be a geodesic sphere of a rank one symmetric space $M_{i}$ of radius $t\in(0,\operatorname{inj}(M_{i}))$. Then, $\mathsf{S}_{1}(t)$ and $\mathsf{S}_{2}(t)$ are isometric if and only if $M_{1}$ and $M_{2}$ are isometric. Thus, by Remark 4.6, two well-positioned totally geodesic submanifolds $\Sigma_{1}$ and $\Sigma_{2}$ of $\mathsf{S}_{t}(p)$ passing through $o\in\mathsf{S}_{t}(p)$ are isotropy-congruent at $o\in\mathsf{S}_{t}(p)$ if and only if they are the intersection of $\mathsf{S}_{t}(p)$ with totally geodesic submanifolds $M_{1}$ and $M_{2}$ of $\bar{M}$ which are isotropy-congruent at $p$. Moreover, observe that $M_{1}$ and $M_{2}$ are isotropy-congruent at $p$ if and only if they are congruent in $\bar{M}$ and contain $p\in\bar{M}$. Hence, by using the classification of totally geodesic submanifolds in symmetric spaces of rank one (see Figure 1) we obtain the classification in Table 1. Consequently, two isometric well-positioned totally geodesic submanifolds of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ passing through $o\in\mathsf{S}^{n}_{\mathbb{F},\tau}$ must be isotropy-congruent at $o$. This implies, by Lemma 3.4, that every well-positioned totally geodesic submanifold of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ is extrinsically homogeneous.

4.1. Examples of well-positioned totally geodesic submanifolds in Hopf-Berger spheres

A good source of examples of well-positioned totally geodesic submanifolds of Hopf-Berger spheres is provided by fixed-point components of isometries as it shows the following lemma.

Lemma 4.8**.**

Let $\Omega$ be a subset of $\operatorname{Isom}(\mathsf{S}^{n}_{\mathbb{F},\tau})$. Then, the set of fixed points of $\Omega$ is a well-positioned totally geodesic submanifold of $\mathsf{S}^{n}_{\mathbb{F},\tau}$.

Proof.

Let us regard the Hopf-Berger spheres as geodesic spheres $\mathsf{S}_{t}(p)$ of an appropiate rank one symmetric space $\bar{M}$. Recall that every isometry of $\mathsf{S}_{t}(p)$ can be identified with a certain isometry of $\bar{M}$ lying in the isotropy at $p\in\bar{M}$, see Remark 2.9. Let $\Omega$ be a subset of isometries of $\bar{M}$ that fix $p$. Consider the totally geodesic submanifold $\bar{\Sigma}$ of $\bar{M}$ given by the connected component by $p$ of the fixed set of $\Omega$. Let $\Sigma\subset\mathsf{S}_{t}(p)$ be the set of fixed points of $\Omega_{|\mathsf{S}_{t}(p)}$, where $\Omega_{|\mathsf{S}_{t}(p)}$ denotes the set of elements in $\Omega$ restricted to $\mathsf{S}_{t}(p)$. Then $\Sigma$ is a totally geodesic submanifold of $\mathsf{S}_{t}(p)$ and $\Sigma=\bar{\Sigma}\cap\mathsf{S}_{t}(p)$. Thus, Theorem 4.4 implies that $\Sigma$ is a well positioned totally geodesic submanifold of $\mathsf{S}_{t}(p)$.

Also notice that by 2.8, $\Sigma$ is connected. In fact, $\Sigma$ can be regarded, using normal coordinates at $p$, as the intersection of $T_{p}\bar{\Sigma}$ with the Euclidean sphere centered at [math] and radius $t$. ∎

It will be convenient for the subsequent discussions to posses explicit descriptions of the tangent space of some well-positioned totally geodesic submanifolds of $\mathsf{S}^{n}_{\mathbb{F},\tau}$. Consider the following subspaces $(1)_{\mathbb{F}}$ and $(2)_{\mathbb{F}}$ of $\mathfrak{p}$, where $\mathbb{F}\in\{\mathbb{C},\mathbb{H},\mathbb{O}\}$.

1. $(1)_{\mathbb{C}}$

$\mathfrak{p}_{\Sigma}=V$, where $V$ is a totally real subspace of $\mathfrak{p}_{2}$. 2. $(2)_{\mathbb{C}}$

$\mathfrak{p}_{\Sigma}=\mathfrak{p}_{1}\oplus V$, where $V\subset\mathfrak{p}_{2}$ is totally complex. 3. $(1)_{\mathbb{H}}$

$\mathfrak{p}_{\Sigma}=\mathbb{R}X\oplus V$, where $X\in\mathfrak{p}_{1}$ and $V$ is a totally complex subspace of $\mathfrak{p}_{2}$ with respect to $J_{X}$, see in Subsection 2.4 for the definition of the map $J$, and also [16, §2.4 (2)]. 4. $(2)_{\mathbb{H}}$

$\mathfrak{p}_{\Sigma}=\mathfrak{p}_{1}\oplus V$, where $V$ is invariant under $J_{X}$ for every $X\in\mathfrak{p}_{1}$. 5. $(1)_{\mathbb{O}}$

$\mathfrak{p}_{\Sigma}=\operatorname{span}\{X_{i}\}_{i\in I}\oplus\mathbb{H}_{I}Y_{1}$, where $I$ is a line of the Fano plane (see Figure 1). 6. $(2)_{\mathbb{O}}$

$\mathfrak{p}_{\Sigma}=\mathfrak{p}_{1}$.

It is clear from Equations (8-13) that these subspaces are invariant under the difference and curvature tensor. Hence, by Lemma 3.1, they induce totally geodesic submanifolds in $\mathsf{S}_{\mathbb{F},\tau}^{n}$. Additionally, it can be checked that these subspaces are invariant under $U$, see Subsection 2.2 for the definition of $U$.

On the one hand, if we consider Example $(1)_{\mathbb{F}}$, for each $\mathbb{F}\in\{\mathbb{C},\mathbb{H},\mathbb{O}\}$, we have that $\mathfrak{k}_{\Sigma}:=[\mathfrak{p}_{\Sigma},\mathfrak{p}_{\Sigma}]_{\mathfrak{k}}$ is isomorphic to $\mathfrak{so}_{k}$, $\mathfrak{u}_{k}$, or $\mathfrak{sp}_{1}\oplus\mathfrak{sp}_{1}$, where $k$ is the real or complex dimension of $V$, respectively. The Lie algebra generated by $\mathfrak{p}_{\Sigma}$ is $\mathfrak{g}_{\Sigma}=\mathfrak{k}_{\Sigma}\oplus\mathfrak{p}_{\Sigma}$, which is isomorphic to $\mathfrak{so}_{k+1}$, $\mathfrak{u}_{k+1}$ or $\mathfrak{sp}_{2}\oplus\mathfrak{sp}_{1}$, respectively. On the other hand, if we consider Example $(2)_{\mathbb{F}}$, we have that $\mathfrak{k}_{\Sigma}$ is isomorphic to $\mathfrak{u}_{k}$, $\mathfrak{sp}_{k}$, or $\mathfrak{so}_{7}$, where $k$ is the complex or quaternionic dimension of $V$, respectively, and the Lie algebra generated by $\mathfrak{p}_{\Sigma}$ is $\mathfrak{g}_{\Sigma}=\mathfrak{k}_{\Sigma}\oplus\mathfrak{p}_{\Sigma}$, which is isomorphic to $\mathfrak{u}_{k+1}$, $\mathfrak{sp}_{k+1}$, or $\mathfrak{so}_{8}$. Let $\mathsf{G}_{\Sigma}$ be the connected subgroup of $\mathsf{G}$ with Lie algebra $\mathfrak{g}_{\Sigma}$. Observe that $\mathfrak{p}_{\Sigma}$ is identified with $T_{o}(\mathsf{G}_{\Sigma}\cdot o)$, and the connected component of the isotropy of $\mathsf{G}_{\Sigma}$ at $o\in\mathsf{S}^{n}_{\mathbb{F},\tau}$ is $\mathsf{K}_{\Sigma}$, which is the connected Lie subgroup of $\mathsf{G}$ with Lie algebra $\mathfrak{k}_{\Sigma}$. Now, using Equations (2) and (3), and the fact that $\mathfrak{p}_{\Sigma}$ is invariant under $D$ and $U$, we have that the second fundamental form of $\Sigma=\mathsf{G}_{\Sigma}\cdot o$ vanishes at $o$. Thus, since $\Sigma$ is an orbit, $\Sigma$ is a totally geodesic submanifold of $\mathsf{S}^{n}_{\mathbb{F},\tau}$, and $\Sigma=\exp_{o}(\mathfrak{p}_{\Sigma})$ by the uniqueness of complete totally geodesic submanifolds. Hence, these totally geodesic submanifolds are extrinsically homogeneous submanifolds.

Moreover, by making use of the Gauss equation, we can compute the curvature of these examples using Equations (8), (9), (10) and (11), (12), (13). It turns out that Example $(1)_{\mathbb{F}}$ is isometric, for each $\mathbb{F}\in\{\mathbb{C},\mathbb{H},\mathbb{O}\}$, to $\mathsf{S}^{k}_{1}$, $\mathsf{S}^{2k+1}_{\mathbb{C},\tau}$, or $\mathsf{S}^{7}_{\mathbb{H},\tau}$, respectively. Furthermore, Example $(2)_{\mathbb{F}}$ is isometric, for each $\mathbb{F}\in\{\mathbb{C},\mathbb{H},\mathbb{O}\}$, to $\mathsf{S}_{\mathbb{C},\tau}^{2k+1}$, $\mathsf{S}_{\mathbb{H},\tau}^{4k+3}$, or $\mathsf{S}^{7}_{1/\tau}$, respectively.

Remark 4.9*.*

Notice that for $\tau\neq 1$ and $n\geq 1$, we have the totally geodesic inclusions

[TABLE]

for the totally geodesic submanifolds constructed in Subsection 4.1.

Lemma 4.10**.**

Let $V$ be a curvature invariant $2$-plane in $\mathsf{S}^{15}_{\mathbb{O},\tau}$ with $\tau\neq 1$. Then, $V$ is contained in the tangent space of a well-positioned totally geodesic $\mathsf{S}^{7}_{\mathbb{H},\tau}$ in $\mathsf{S}^{15}_{\mathbb{O},\tau}$.

Proof.

First of all, using the isotropy of $\mathsf{S}^{15}_{\mathbb{O},\tau}$ (see [48]), we can assume that the $2$-plane $V$ is spanned by $u,v\in\mathfrak{p}$, where

[TABLE]

where we are using the notation of Subsection 2.4. Let us proceed by contradiction. Then, we can assume that $b_{4}\neq 0$, since otherwise, $V$ is contained in the tangent space of some well-positioned totally geodesic submanifold $\mathsf{S}^{7}_{\mathbb{H},\tau}$ of $\mathsf{S}^{15}_{\mathbb{O},\tau}$, see example $(1)_{\mathbb{O}}$ above. Now,

[TABLE]

since $V$ is curvature invariant. Thus we either have that $b_{6}=0$ or $a_{1}=a_{4}=0$.

Let us assume that $b_{6}=0$. Thus,

[TABLE]

since $V$ is curvature invariant. Hence, $a_{1}=a_{2}=a_{3}=0$ or $b_{5}=0$. However, in the first case, $V$ is contained in the tangent space of some well-positioned totally geodesic submanifold $\mathsf{S}^{7}_{\mathbb{H},\tau}$ of $\mathsf{S}^{15}_{\mathbb{O},\tau}$. Thus, $b_{5}=0$. Observe that we can assume that $a_{4}\neq 0$, since otherwise $V$ is contained in the tangent space of some totally geodesic submanifold $\mathsf{S}^{7}_{\mathbb{H},\tau}$ of $\mathsf{S}^{15}_{\mathbb{O},\tau}$. Then,

[TABLE]

since $V$ is curvature invariant. Hence, $a_{1}=a_{2}=a_{3}=0$ and $V$ is contained in the tangent space of some well-positioned totally geodesic submanifold $\mathsf{S}^{7}_{\mathbb{H},\tau}$ of $\mathsf{S}^{15}_{\mathbb{O},\tau}$.

Let us assume that $b_{6}\neq 0$. Then, by Equation (14), $a_{1}=a_{4}=0$. Moreover,

[TABLE]

since $V$ is curvature invariant. Hence, $a_{2}=a_{3}=0$, implying that $V$ is contained in the tangent space of some well-positioned totally geodesic submanifold $\mathsf{S}^{7}_{\mathbb{H},\tau}$ of $\mathsf{S}^{15}_{\mathbb{O},\tau}$.∎

5. Not well-positioned totally geodesic submanifolds

The purpose of this section is to study not well-positioned totally geodesic submanifolds. We start by studying the $1$-dimensional case, i.e. geodesics. Subsequently, we prove Theorem 5.7, which gives a characterization of not well-positioned totally geodesic submanifolds in Hopf-Berger spheres. As a consequence of these and other previous results we are able to give the proof of Theorem A. Finally, we prove Theorem 5.10, which provides a description of not well-positioned maximal totally geodesic spheres in Hopf-Berger spheres.

5.1. Geodesics of Hopf-Berger spheres

In what follows we will focus on $1$-dimensional totally geodesic submanifolds of Hopf-Berger spheres, i.e. geodesics.

Lemma 5.1**.**

Every flat totally geodesic submanifold of a Hopf-Berger sphere $\mathsf{S}^{n}_{\mathbb{F},\tau}$ is a geodesic.

Proof.

Let us recall that $\mathsf{K}$ acts polarly on $T_{o}\mathsf{S}^{n}_{\mathbb{F},\tau}$ (see Remark 2.10). Moreover, the principal $\mathsf{K}$-orbits are products of spheres and they have codimension two. Let $\Sigma^{\prime}$ be a flat totally geodesic submanifold of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ of dimension greater than or equal two and let $\Sigma$ be a totally geodesic (and hence flat) surface of $\Sigma^{\prime}$. We may assume that $o\in\Sigma$. Then, by Remark 3.3, $T_{o}\Sigma$ is a section for the polar representation of $\mathsf{K}$ in $T_{o}\mathsf{S}^{n}_{\mathbb{F},\tau}$, since $\mathsf{K}$ acts with cohomogeneity two. Hence, $T_{o}\Sigma$ is spanned by a horizontal and a vertical vector, and thus $\Sigma$ is well-positioned at $o$. By Lemma 4.2, $\Sigma$ is well-positioned. Then, by Theorem 4.4 and Remark 4.6, $\Sigma$ is a geodesic sphere of a totally geodesic submanifold of dimension three of the rank one symmetric space containing $\mathsf{S}^{n}_{\mathbb{F},\tau}$ as a geodesic sphere. Thus, $\Sigma$ is a round sphere of dimension two, and therefore we obtain a contradiction with the fact that $\Sigma$ is flat. ∎

Lemma 5.2**.**

Let $\tau\neq 1$ and $\gamma$ be a geodesic in $\mathsf{S}^{n}_{\mathbb{F},\tau}=\mathsf{G}/\mathsf{K}$. Then, there exists a totally geodesic submanifold $\Sigma$ of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ satisfying that:

1. i)

$\gamma$* is contained in $\Sigma$.* 2. ii)

$\Sigma$* is well-positioned.* 3. iii)

$\Sigma$* is isometric to $\mathsf{S}^{3}_{\mathbb{C},\tau}$.* 4. iv)

$\Sigma$* is an orbit of a subgroup of $\operatorname{Isom}(\mathsf{S}^{n}_{\mathbb{F},\tau})$ locally isomorphic to $\mathsf{U}_{2}$.*

Proof.

Let us assume that $\gamma$ is a (non-constant) geodesic of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ with initial velocity $v_{1}+v_{2}\in T_{o}\mathsf{S}^{n}_{\mathbb{F},\tau}$, where $v_{1}\in\mathfrak{p}_{1}$ and $v_{2}\in\mathfrak{p}_{2}$. If $v_{1}$ or $v_{2}$ is equal to zero, let us take any (not well-positioned) geodesic $\widetilde{\gamma}$ whose velocity has horizontal or vertical projection equal to $v_{2}$ or $v_{1}$, respectively. Hence, if $\widetilde{\gamma}$ is contained in a well-positioned totally geodesic submanifold $\Sigma$, then $\gamma$ is also contained in $\Sigma$. Hence, we may assume that $v_{1}$ and $v_{2}$ are both not zero. Then, the initial velocity of $\gamma$ is contained in the subspace spanned by $\{v_{1},v_{2},[v_{1},v_{2}]\}$. This subspace is clearly the tangent space of a well-positioned totally geodesic submanifold $\Sigma$ isometric to $\mathsf{S}^{3}_{\mathbb{C},\tau}$, see Subsection 4.1 $(2)_{\mathbb{C}}$ or $(2)_{\mathbb{H}}$, and Remark 4.9. Moreover, by the discussion in Subsection 4.1, we have that $\Sigma$ is an orbit of a subgroup of $\operatorname{Isom}(\mathsf{S}^{n}_{\mathbb{F},\tau})$ locally isomorphic to $\mathsf{U}_{2}$.∎

We define the slope of a geodesic $\gamma$ in $\mathsf{S}^{n}_{\mathbb{F},\tau}$ to be the quotient between the lengths of the vertical and horizontal projections of the velocity of $\gamma$. This quantity is well defined for every geodesic since Hopf-Berger spheres are geodesic orbit spaces, see [45]. Moreover, using the full isotropy representation of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ described in Remark 2.10, it is easy to see that two geodesics are congruent in $\mathsf{S}^{n}_{\mathbb{F},\tau}$ if and only if they have the same slope.

Lemma 5.3**.**

Let $\mathsf{S}^{n}_{\mathbb{F},\tau}$, with $\tau\neq 1$, and let $\gamma$ be a closed geodesic in $\mathsf{S}^{n}_{\mathbb{F},\tau}$. Then, the set of possible slopes for $\gamma$ is countable.

Proof.

First of all, by Lemma 5.2, we can assume that $\gamma$ is a closed geodesic of $\mathsf{S}^{3}_{\mathbb{C},\tau}$ with $\tau\neq 1$. Let us define a naturally reductive decomposition for $\mathsf{S}^{3}_{\mathbb{C},\tau}=\mathsf{U}_{2}/\mathsf{U}_{1}$. Keeping the notation in Subsection 2.4 applied to $\mathsf{S}^{3}_{\mathbb{F},\tau}$, we define $\mathfrak{p}^{\tau}_{1}$ as the one-dimensional subspace of $\mathfrak{g}=\mathfrak{u}_{2}$ spanned by the unit vector

[TABLE]

Hence, it can be checked that $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}^{\tau}$, where $\mathfrak{p}^{\tau}:=\mathfrak{p}^{\tau}_{1}\oplus\mathfrak{p}_{2}$, is a naturally reductive decomposition of $\mathsf{S}^{3}_{\mathbb{C},\tau}$, for each $\tau\in(0,\infty)$. In particular, every geodesic of $\mathsf{S}^{3}_{\mathbb{C},\tau}$ is of the form $\operatorname{Exp}(sX)\cdot o$, for some $X\in\mathfrak{p}^{\tau}$. For our purposes, we can assume that $o=(0,1)\in\mathsf{S}^{3}_{\mathbb{C},\tau}\subset\mathbb{C}^{2}$. Let $X:=\alpha_{1}Y_{1}+\alpha_{2}J_{1}Y_{1}+\alpha_{3}X^{\tau}$, where $\alpha_{i}\in\mathbb{R}$ for every $i\in\{1,2,3\}$ satisfy $\sum_{i=1}^{3}\alpha^{2}_{i}=1$, and consider the geodesic $\gamma_{X}(s)=\operatorname{Exp}(sX)\cdot o$. We define the quantities

[TABLE]

A direct computation shows that

[TABLE]

If $X$ is vertical, $\alpha_{3}=\pm 1$, $\alpha_{1}=\alpha_{2}=0$, and we have $\gamma_{X}(s)=(0,e^{\pm\frac{is}{\sqrt{\tau}}})$. If $X$ is horizontal, $\alpha_{3}=0$, $\alpha_{1}^{2}+\alpha_{2}^{2}=1$ and we have

[TABLE]

Now, let us assume that $X$ has non-trivial projection onto $\mathfrak{p}^{\tau}_{1}$ and $\mathfrak{p}_{2}$. Then, if $\gamma_{X}$ is closed,

[TABLE]

Hence, $\alpha_{3}=\frac{P\sqrt{\tau}(j+2k)}{j(1-\tau)}=\frac{\sqrt{(1-\alpha_{3}^{2})+\tau\alpha_{3}^{2}}\sqrt{\tau}(j+2k)}{j(1-\tau)}$, where $j\in\mathbb{Z}\setminus\{0\}$ and $k\in\mathbb{Z}$. Since the slope of $\gamma_{X}$ is given by $\sqrt{\tfrac{\alpha^{2}_{3}}{1-\alpha^{2}_{3}}}$, we deduce that the set of possible slopes for $\gamma_{X}$ is countable.∎

5.2. A characterization of not well-positioned totally geodesic submanifolds

Let us start by proving that all the geodesics of a not well-positioned totally geodesic submanifold are closed, have the same slope, and thus, the same length.

Lemma 5.4**.**

Let $f\colon\Sigma\rightarrow\mathsf{S}^{n}_{\mathbb{F},\tau}$ be a not well-positioned totally geodesic immersion of a complete Riemannian manifold $\Sigma$ with dimension $d\geq 2$. Then, the following statements hold:

1. i)

$\Sigma$* is compact.* 2. ii)

Every geodesic of $\Sigma$ is closed. 3. iii)

The slope of $f_{*}(v)$ does not depend on $v\in T\Sigma\setminus\{0\}$.

Proof.

Firstly, i) is a consequence of Lemma 3.2 and Lemma 5.1.

Let $\gamma_{v}$ be the geodesic of $\Sigma$ with initial conditions $q\in\Sigma$ and $v\in T_{q}\Sigma$. Also, consider the geodesic $\tilde{\gamma}=f\circ\gamma_{v}$ of $\mathsf{S}^{n}_{\mathbb{F},\tau}$. By Proposition 5.2, there exists an embeded and compact totally geodesic submanifold $N$ of $\mathsf{S}^{n}_{\mathbb{F},\tau}$, which is isometric to $\mathsf{S}^{3}_{\mathbb{C},\tau}$, such that $\tilde{\gamma}(\mathbb{R})\subset N$. Since $\Sigma$ and $N$ are compact it is not difficult to show that $f(F)=\tilde{F}$, where $F$ and $\tilde{F}$ are the closure of $\gamma_{v}(\mathbb{R})$ and $\tilde{\gamma}(\mathbb{R})$, respectively. Observe that $\tilde{F}\subset N$.

Since $\Sigma$ and $N$ are compact g.o. spaces, then $F$ and $\tilde{F}$ are the orbits of abelian compact Lie subgroups $\mathsf{A}$, $\tilde{\mathsf{A}}$ of the full isometry groups of $\Sigma$ and $N$, respectively. In particular, $F$ and $\tilde{F}$ are compact submanifolds of $\Sigma$ and $N$, respectively. Moreover, since $f$ is an immersion, $f\colon F\to\tilde{F}$ is a covering map. Asumme that $\gamma_{v}$ is not closed. Then $\dim(\tilde{F})=\dim(F)\geq 2$. Thus, the subspace $f_{*q}(T_{q}F)=T_{f(q)}\tilde{F}\subset T_{f(q)}N$ intersects non-trivially the horizontal $2$-dimensional subspace of $N$, since $\dim(N)=3$. Consequently, by Lemma 4.2 and Lemma 4.1, $f(\Sigma)$ is well positioned, yielding a contradiction. This proves ii).

Observe that $f\circ\gamma_{v}$ is a closed geodesic for all $v\in T\Sigma$. Consequently, by Lemma  5.3, we can prove using a continuity argument that the slope of $f_{*q}(v)$ does not depend on $v\neq 0$. ∎

Lemma 5.5**.**

Let $\Sigma$ be a not well-positioned totally geodesic submanifold of $\mathsf{S}^{n}_{\mathbb{F},\tau}=\mathsf{G}/\mathsf{K}$ with dimension $d\geq 2$. Then, $\Sigma$ is isometric to a compact rank one symmetric space.

Proof.

Let us assume that $o\in\Sigma$ and let $\mathsf{S}(T_{o}\Sigma)$ be the unit sphere of $T_{o}\Sigma$. Every vector in $T_{o}\Sigma$ has the same slope by Lemma 5.4. Let $v\in\mathsf{S}(T_{o}\Sigma)$, then $\mathsf{K}\cdot v$ contains $\mathsf{S}(T_{o}\Sigma)$, see Remark 2.10. Let $w\in T_{v}(\mathsf{S}(T_{o}\Sigma))$. Then, there exists an element $X\in\mathfrak{k}\subset\mathfrak{so}(T_{o}\mathsf{S}^{n}_{\mathbb{F},\tau})$ such that $Xv=w$. Let ${\mathfrak{k}}_{\Sigma}$ be the Lie algebra of the full isotropy group of $\Sigma$ at $o$, and $X_{\Sigma}$ be the projection of the restriction of $X$ to $T_{o}\Sigma$. Then $X_{\Sigma}\in\mathfrak{k}_{\Sigma}$. Moreover, $X_{\Sigma}v=w$. This implies that the isotropy of $\Sigma$ acts transitively on the unit sphere of its tangent space at $o$. Therefore, since $\Sigma$ is a compact homogeneous space, $\Sigma$ is a compact $2$-point homogeneous space. Consequently, $\Sigma$ is isometric to a compact rank one symmetric space, see [44].∎

Remark 5.6*.*

Notice that if $\Sigma$ is embedded, the proof of Lemma 5.5 follows by a celebrated result in the theory of manifolds with all the geodesics closed. This result asserts that a compact homogeneous space with all its geodesics closed and of the same length is isometric to a compact rank one symmetric space, see [10, Theorem 7.55, p. 196]. Indeed, by Lemma 5.4, $\Sigma$ is a compact manifold with all its geodesics closed and of the same length, since they all have the same slope. Moreover, since $\Sigma$ is a totally geodesic submanifold of a homogeneous space is intrinsically homogeneous. Then, $\Sigma$ is isometric to a compact rank one symmetric space.

Recall that every Hopf-Berger sphere can be regarded, up to a homothety, as a geodesic sphere of an appropriate rank one symmetric space $\bar{M}$. Let $\mathsf{S}_{t}(p)$ be the geodesic sphere of $\bar{M}$ with center $p$ and radius $0<t<\operatorname{inj}(p)$. Observe that $\operatorname{inj}(p)=\pi/2$ if $\bar{M}$ is of compact type and $\infty$ if $\bar{M}$ is of non-compact type. Let $I\!I^{t}$ be the second fundamental form of $\mathsf{S}_{t}(p)$ and $\mathcal{S}^{t}$ the shape operator of $\mathsf{S}_{t}(p)\subset\bar{M}$. By Remark 2.6, we have

[TABLE]

where $\beta_{1}^{t}$ and $\beta_{2}^{t}$ are scalars defined by equations (5) and (6), respectively. From these expressions one obtains that $\beta_{1}^{t}\neq\beta_{2}^{t}$.

We define a symmetric bilinear form $\upalpha$ on $\mathsf{S}_{t}(p)$ given by

[TABLE]

where $\xi$ is the outer unit normal field to the geodesic sphere. Observe that $\upalpha$ is positive definite if and only if $t\in(0,\tfrac{\pi}{4})$ or $t>1$, see Equations (5) and (6). By identifying $\mathsf{S}_{t}(p)$ with the appropiate Hopf-Berger sphere $\mathsf{S}^{n}_{\mathbb{F},\tau}$, we can equip $\mathsf{S}^{n}_{\mathbb{F},\tau}$ with $\upalpha$. Consequently, $\upalpha$ is non-degenerate for $\tau\neq\frac{1}{2}$. Moreover, $\upalpha$ defines a Riemannian metric if and only if $\tau>\tfrac{1}{2}$, and a pseudo-Riemannian metric with index equal to $\dim\mathbb{F}-1$ for $\tau<\tfrac{1}{2}$, where $\mathbb{F}\in\{\mathbb{C},\mathbb{H},\mathbb{O}\}$.

We say that a vector $v\in T_{p}\mathsf{S}^{n}_{\mathbb{F},\tau}$ is $\upalpha$-isotropic if $\upalpha(v,v)=0$, and a subspace $V$ of $T_{p}\mathsf{S}^{n}_{\mathbb{F},\tau}$ is $\upalpha$-isotropic if every vector $v\in V$ is $\upalpha$-isotropic.

Theorem 5.7**.**

Let $\Sigma$ be a complete immersed submanifold of $\mathsf{S}^{n}_{\mathbb{F},\tau}$, $\tau\neq 1$, with dimension $d\geq 2$, and let $\bar{M}$ be the rank one symmetric space containing $\mathsf{S}^{n}_{\mathbb{F},\tau}$ as a geodesic sphere. If $\tau\neq\tfrac{1}{2}$, then the following statements are equivalent:

1. i)

$\Sigma$* is a not well-positioned totally geodesic submanifold of $\mathsf{S}^{n}_{\mathbb{F},\tau}$.* 2. ii)

$T_{p}\Sigma$* is $\upalpha$-isotropic and invariant under the curvature tensor of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ for every $p\in\Sigma$.*

Moreover, under any of the above conditions $\Sigma$ is a totally geodesic submanifold of $\bar{M}$, and if $\tau=\tfrac{1}{2}$, every totally geodesic submanifold of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ is well-positioned.

Proof.

Let us prove that a not well-positioned totally geodesic submanifold ${\Sigma}$ of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ with dimension $d\geq 2$ is $\upalpha$-isotropic. Firstly, by Lemma 5.5, we have that ${\Sigma}$ is isometric to a compact rank one symmetric space. Thus, by the classification of totally geodesic submanifolds in rank one symmetric spaces, $\Sigma$ contains a totally geodesic surface $\widetilde{\Sigma}$, which is a not well-positioned totally geodesic submanifold of $\mathsf{S}^{n}_{\mathbb{F},\tau}$, see Figure 4. By Corollary 4.5, we have that $\mathbb{F}=\mathbb{H}$ or $\mathbb{F}=\mathbb{O}$. Moreover, by Lemma 4.10, we can assume without loss of generality that $\widetilde{\Sigma}$ is a not well-positioned totally geodesic submanifold of $\mathsf{S}^{4m+3}_{\mathbb{H},\tau}$.

Let $\mathfrak{p}_{\widetilde{\Sigma}}$ be the tangent space of $\widetilde{\Sigma}$ at $o\in\widetilde{\Sigma}$ identified with a subspace of the reductive complement $\mathfrak{p}$ of $\mathsf{S}^{4m+3}_{\mathbb{H},\tau}$, see Subsection 2.4. Using the isotropy of $\mathsf{S}^{4m+3}_{\mathbb{H},\tau}$ (see [48]), we can assume that $m=2$, and $\mathfrak{p}_{\widetilde{\Sigma}}$ is spanned by the basis $\{u,v\}$ given by

[TABLE]

where $a_{i},b_{j}\in\mathbb{R}$ for $i\in\{1,\ldots,4\}$ and $j\in\{1,\ldots,6\}$.

By Lemma 4.1, $\mathfrak{p}_{\widetilde{\Sigma}}$ does not contain horizontal or vertical vectors, since $\widetilde{\Sigma}$ is not well-positioned. Hence, $a_{4}\neq 0$. Moreover, we can assume without loss of generality that $b_{4}=0$. Otherwise, the set $\{u,v-b_{4}/a_{4}u\}$ is also a basis for $\mathfrak{p}_{\widetilde{\Sigma}}$ where the second vector projects trivially over $Y_{1}$.

Since $\mathfrak{p}_{\widetilde{\Sigma}}$ is curvature invariant, we have

[TABLE]

Every totally geodesic submanifold of a homogeneous space is intrinsically homogeneous and every homogeneous space of dimension two has constant sectional curvature. Then, every covariant derivative of the curvature tensor $R$ restricted to $\mathfrak{p}_{\widetilde{\Sigma}}$ vanishes. Thus,

[TABLE]

This implies that $a_{1}=0$ or $b_{5}=0$. In the latter case, $b_{6}\neq 0$, since otherwise there would be a vertical vector in $\mathfrak{p}_{\widetilde{\Sigma}}$. Thus, by Equation (17)

[TABLE]

Hence, $(a_{1},a_{2},a_{3})$ and $(b_{1},b_{2},b_{3})$ are proportional and there would be a horizontal vector in $\mathfrak{p}_{\widetilde{\Sigma}}$. Now assume that $b_{5}\neq 0$, and thus $a_{1}=0$. Then,

[TABLE]

Thus, $b_{1}=0$ or $a_{2}=a_{3}=0$, but the latter cannot happen because it would imply the existence of a horizontal vector. Hence, $b_{1}=0$. Moreover, we have

[TABLE]

This yields $\tau=1/2$ or $a_{2}b_{2}+a_{3}b_{3}=0$. Let us assume that $\tau=1/2$. Then,

[TABLE]

Hence, $b_{2}=b_{3}=0$ and we obtain a contradiction due to the existence of a horizontal vector in $\mathfrak{p}_{\widetilde{\Sigma}}$. This proves that there are no not well-positioned totally geodesic surfaces in $\mathsf{S}^{n}_{\mathbb{F},1/2}$.

Now, let us assume that $a_{2}b_{2}+a_{3}b_{3}=0$. Then, there exist $r,s\in\mathbb{R}$ and $\varphi\in[0,2\pi]$ such that

[TABLE]

Thus,

[TABLE]

Hence,

[TABLE]

To sum up, we have

[TABLE]

Now a simple computation yields that $\mathfrak{p}_{{\widetilde{\Sigma}}}$ is $\upalpha$-isotropic. Therefore, by Lemma 5.4, ${\Sigma}$ is $\upalpha$-isotropic. This completes the proof that i) implies ii) and since there are no not well-positioned totally geodesic surfaces in $\mathsf{S}^{n}_{\mathbb{F},1/2}$, we also have proved that there are no not well-positioned totally geodesic submanifolds in $\mathsf{S}^{n}_{\mathbb{F},1/2}$.

Conversely, by the Gauss equation associated with the embedding of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ in $\bar{M}$, if $T_{o}\Sigma$ is $\upalpha$-isotropic and invariant under the curvature tensor of $\mathsf{S}^{n}_{\mathbb{F},\tau}$, then $T_{o}\Sigma$ is invariant under the curvature tensor of $\bar{M}$. Thus, $\bar{\Sigma}:=\overline{\exp}_{o}(T_{o}\Sigma)$ is a totally geodesic submanifold of $\bar{M}$. Observe that the velocity of every geodesic $\gamma$ of $\Sigma$ is $\upalpha$-isotropic for every $t\in\mathbb{R}$, since $\Sigma$ is a g.o. space. This implies that $\gamma$ is a geodesic of $\bar{\Sigma}$, and the Riemannian exponential map $\exp_{o}$ of $\mathsf{S}^{n}_{\mathbb{F},\tau}$, and the Riemannian exponential map $\overline{\exp}_{o}$ of $\bar{M}$ agree on $T_{o}\Sigma$. Thus, $\Sigma=\exp_{o}(T_{o}\Sigma)=\overline{\exp}_{o}(T_{o}\Sigma)=\bar{\Sigma}$. Consequently, $\Sigma$ is a totally geodesic submanifold of $\bar{M}$.∎

By Remark 4.6, well positioned totally geodesic submanifolds of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ are embedded. Moreover, by Theorem 5.7, every not well-positioned totally geodesic submanifold is also embedded. Then, we have the following result.

Corollary 5.8**.**

Let $\Sigma$ be a complete totally geodesic submanifold of $\mathsf{S}^{n}_{\mathbb{F},\tau}$. Then, $\Sigma$ is embedded in $\mathsf{S}^{n}_{\mathbb{F},\tau}$.

Now we are in a position to prove the first main result of this article.

Proof of Theorem A.

Notice that $\upalpha$-isotropic subspaces exist if and only if $\tau\leq\tfrac{1}{2}$, see Equations (5), (6), and (16). Then, by Theorem 5.7, if $\tau\geq\tfrac{1}{2}$ we have that every totally geodesic submanifold of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ is well-positioned. Also, by Corollary 4.5, if $\dim\Sigma\geq\dim\mathbb{F}$, then $\Sigma$ is well-positioned.

Let us identify $\mathsf{S}^{n}_{\mathbb{F},\tau}$ with a geodesic sphere $\mathsf{S}(p)$ centered at some point $p$ of a rank one symmetric space $\bar{M}$. The intersection of a geodesic sphere $\mathsf{S}(p)$ of a symmetric space of rank one $\bar{M}$ with a complete totally geodesic submanifold $M$ passing through $p\in\bar{M}$ is a geodesic sphere of $M$ (see Remark 4.6). Hence, the intersection of $\mathsf{S}(p)$ with a complete totally geodesic submanifold $M$ of $\bar{M}$ of dimension $d^{\prime}\geq 3$ containing $p$ has dimension $d^{\prime}-1$. Thus, it follows by Theorem 4.4 that i) and ii) are equivalent.

Finally, by Remark 4.7, two isometric well-positioned totally geodesic submanifold must be congruent, and this concludes the proof. ∎

5.3. Not well-positioned maximal totally geodesic spheres

This subsection is devoted to the study of not well-positioned maximal totally geodesic spheres. Thus, by Theorem A, we may assume that $\tau<1/2$, and $\mathbb{F}=\mathbb{H}$ or $\mathbb{F}=\mathbb{O}$.

Recall that we have the following presentations: $\mathsf{S}^{4n+3}_{\mathbb{H},\tau}=\mathsf{G}/\mathsf{K}=\mathsf{Sp}_{n+1}/\mathsf{Sp}_{n}$ and $\mathsf{S}^{15}_{\mathbb{O},\tau}=\mathsf{G}/\mathsf{K}=\mathsf{Spin}_{9}/\mathsf{Spin}_{7}$. Let $o\in\mathsf{S}^{n}_{\mathbb{F},\tau}$ be fixed and let $F(o)$ be the fiber of the corresponding Hopf fibration by $o$. Let us consider the reductive decomposition of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ appearing in Subsection 2.4, which is given by $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$, with $\mathfrak{p}=\mathfrak{p}_{1}\oplus\mathfrak{p}_{2}$, where $\mathfrak{p}_{1}$ is identified with the vertical distribution $\mathcal{V}$ at $o$, i.e. the tangent space of the fiber $F(o)$ at $o\in\mathsf{S}^{n}_{\mathbb{F},\tau}$, and $\mathfrak{p}_{2}$ is identified with the horizontal distribution $\mathcal{H}$ at $o\in\mathsf{S}^{n}_{\mathbb{F},\tau}$.

Let $X\in\mathfrak{p}_{2}$ be a unit vector. In order to simplify our computations we will assume that $X=Y_{1}$, defined as in Subsection 2.4. Using Equation (2), we have for each $i\in\{1,\ldots,\dim\mathbb{F}-1\}$,

[TABLE]

Thus, Equation (18) implies that the linear transformation $e^{s(\nabla Y_{1}^{*})_{o}}$ of $\mathfrak{p}$ is the rotation of angle $s\sqrt{\tau}$ in each $2$-plane spanned by $\{(J_{i}Y_{1})^{*},X^{*}_{i}\}$, and fixes the orthogonal complement of all these $2$-planes. Hence, in particular, $e^{s(\nabla Y_{1}^{*})_{o}}$ fixes $Y_{1}$. Thus, since the slope of the subspace $e^{s(\nabla Y_{1}^{*})_{o}}\mathcal{V}_{o}$, which is spanned by $\{e^{s(\nabla Y_{1}^{*})_{o}}(X_{i})^{*}_{o}\}^{\dim\mathbb{F}-1}_{i=1}$, is [math] for $s=0$, and is $\infty$ for $s=\frac{\pi}{2\sqrt{\tau}}$, by continuity, the subspace $e^{s^{\prime}(\nabla Y_{1}^{*})_{o}}\mathcal{V}_{o}$ is $\upalpha$-isotropic for some $s^{\prime}>0$.

Let us denote by $\phi_{s}$ the flow associated with $Y_{1}^{*}$ in $\mathsf{S}^{n}_{\mathbb{F},\tau}$. Then, the integral curve of $Y_{1}^{*}$ given by $\phi_{s}(o)=\gamma(s)$ is a geodesic of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ by Equation (18). Moreover, let us consider the Equation (2.2.1) of  [17], which relates the parallel transport $\mathsf{P}_{\gamma}$ along a geodesic $\gamma$ with the pushforward $\phi_{s_{*}}$ of the flow of $Y_{1}^{*}$ in the following manner

[TABLE]

In the following, we regard, up to a homothetical factor, the Hopf-Berger sphere $\mathsf{S}^{n}_{\mathbb{F},\tau}$ of factor $\tau<1/2$ as a geodesic sphere $\mathsf{S}_{t}(p)$ of radius $t\in(\pi/4,\pi/2)$ of a symmetric space $\bar{M}$ of sectional curvatures between $1$ and $4$, see Equation (7). The shape operator $\mathcal{S}^{t}$ of the geodesic sphere $\mathsf{S}_{t}(p)$ with respect to the outer unit normal vector field $\xi$ leaves the vertical and horizontal distributions invariant and has only two eigenvalues of different sign: the vertical one $\beta_{1}^{t}=-2\cot(2t)$ and the horizontal one $\beta_{2}^{t}=-\cot(t)$, see Remark 2.6. Let $\bar{\nabla}$ and $\bar{R}$ be the Levi-Civita connection and the curvature tensor of $\bar{M}$. We have $\langle\bar{\nabla}_{(Y_{1})^{*}_{o}}(Y_{1})^{*}_{o},\xi_{o}\rangle=-\langle\mathcal{S}^{t}(Y_{1})^{*}_{o}),(Y_{1})^{*}_{o}\rangle=\cot(t)=\sqrt{\frac{\tau}{1-\tau}}$. Then,

[TABLE]

The tangent space of a Helgason sphere $\mathsf{S}_{4}^{\dim\mathbb{F}}$ that contains the radial geodesic from the center $p$ of $\mathsf{S}_{t}(p)$ to $o\in\mathsf{S}_{t}(p)$ is $T_{o}\mathsf{S}_{4}^{\dim\mathbb{F}}=\mathcal{V}_{o}\oplus\mathbb{R}\xi_{o}$. Hence, $T_{o}\mathsf{S}_{4}^{\dim\mathbb{F}}$ is invariant under $\bar{R}$, since $\mathsf{S}_{4}^{\dim\mathbb{F}}$ is a totally geodesic submanifold of $\bar{M}$.

Recall that the isometries of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ can be extended to isometries in the isotropy of $\bar{M}$ at $p\in\bar{M}$, see Remark 2.9. Hence, for simplicity, we will also denote by $(Y_{1})^{*}$ and $\phi_{s}$, the Killing vector field induced by $Y_{1}$ in $\bar{M}$ and its associated flow, respectively. Then, taking into account that the flow $\phi_{s}$ is given by isometries, we deduce that $\mathrm{(}{\phi}_{s})_{*o}(T_{o}\mathsf{S}_{4}^{\dim\mathbb{F}})$ and $({\phi}_{s})_{*o}(\mathcal{V}_{o})=\mathcal{V}_{\gamma(s)}\subset T_{\gamma(s)}\mathsf{S}_{t}(p)$ are totally geodesic subspaces of $T_{\gamma(s)}\bar{M}$. Since $({\phi}_{s})_{*o}(\mathcal{V}_{o})$ is perpendicular to $\gamma^{\prime}(s)$, taking into account that $\gamma(s)$ is a geodesic of $\mathsf{S}_{t}(p)$ and that $\gamma^{\prime}(s)$ is a horizontal eigenvector of $\mathcal{S}^{t}$ for all $s$, one has that

[TABLE]

where $\bar{\mathsf{P}}_{\gamma(s)}$ denotes the parallel transport in $\bar{M}$ along $\gamma(s)=\phi_{s}(o)$. Then, by Equation (19),

[TABLE]

Therefore, $\mathrm{e}^{s({\nabla}Y_{1}^{*})_{o}}\mathcal{V}_{o}$ is the tangent space of a totally geodesic hypersphere of a Helgason sphere of $\bar{M}$, and thus invariant under $\bar{R}$. This implies by Gauss equation that $\mathrm{e}^{s({\nabla}Y_{1}^{*})_{o}}\mathcal{V}_{o}$ is a $R$-invariant subspace of $\mathfrak{p}$. Consequently, taking into account that the Levi-Civita connection is preserved by homotheties, and the paragraph below Equation (18), there is some $s^{\prime}>0$ such that $\mathrm{e}^{s^{\prime}({\nabla}Y_{1}^{*})_{o}}\mathcal{V}_{o}$ is an $\upalpha$-isotropic, $R$-invariant subspace of $\mathfrak{p}$. Therefore, by Theorem 5.7, $\mathrm{e}^{s^{\prime}({\nabla}Y_{1}^{*})_{o}}\mathcal{V}_{o}$ is the tangent space of a not well-positioned totally geodesic submanifold $\widehat{\Sigma}$ of $\mathsf{S}^{n}_{\mathbb{F},\tau}$. Furthermore notice that by Equation (20),

[TABLE]

Thus, this subspace is tangent to a Helgason sphere of $\bar{M}$ and contains the tangent space of $\widehat{\Sigma}$ at $o$ which is equal to $\mathrm{e}^{s^{\prime}(\bar{\nabla}Y_{1}^{*})_{o}}\mathcal{V}_{o}$. Consequently, $\widehat{\Sigma}$ is a totally geodesic hypersurface of the aforementioned Helgason sphere of $\bar{M}$, and by using Equation (18), and the fact that $\mathrm{e}^{s^{\prime}(\bar{\nabla}Y_{1}^{*})_{o}}\mathcal{V}_{o}$ is $\upalpha$-isotropic, we can compute that

[TABLE]

Furthermore, $\mathrm{e}^{s^{\prime}(\bar{\nabla}Y_{1}^{*})_{o}}T_{o}\mathsf{S}_{4}^{\dim\mathbb{F}}$ projects non trivially over $\xi_{o}$. Thus, the Helgason sphere whose tangent space is equal to $\mathrm{e}^{s^{\prime}(\bar{\nabla}Y_{1}^{*})_{o}}T_{o}\mathsf{S}_{4}^{\dim\mathbb{F}}$ and the geodesic sphere $\mathsf{S}_{t}(p)$ intersect transversally at $o$.

Remark 5.9*.*

Observe that if $\mathbb{F}=\mathbb{H}$, by looking at Equation (22), it is clear that $\widehat{\Sigma}$ is totally geodesic of a well-positioned totally geodesic submanifold $\mathsf{S}^{7}_{\mathbb{H},\tau}$ of $\mathsf{S}^{4n+3}_{\mathbb{H},\tau}$, see Subsection 4.1. Furthermore, the totally geodesic submanifold $\widehat{\Sigma}$ is a maximal totally geodesic submanifold of $\mathsf{S}^{7}_{\mathbb{H},\tau}$ and $\mathsf{S}^{15}_{\mathbb{O},\tau}$. Indeed, by Corollary 4.5, $\widehat{\Sigma}$ has the highest dimension that a not well-positioned totally geodesic submanifold can attain, and by Theorem 4.4, well-positioned totally geodesic submanifolds of $\mathsf{S}^{7}_{\mathbb{H},\tau}$ or $\mathsf{S}^{15}_{\mathbb{O},\tau}$ have dimension less or equal than $3$ or $7$, respectively; see Table 1.

Theorem 5.10**.**

Let $\Sigma$ be a complete not well-positioned totally geodesic sphere of $\mathsf{S}^{n}_{\mathbb{F},\tau}$, $\tau\neq 1$. Then, the following statements hold:

1. i)

There is a unique maximal not well-positioned totally geodesic submanifold $\widehat{\Sigma}$ of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ isometric to $\mathsf{S}^{\dim\mathbb{F}-1}_{4-4\tau}$ with $T_{o}\Sigma\subset T_{o}\widehat{\Sigma}$. 2. ii)

Every two not well-positioned totally geodesic spheres in $\mathsf{S}^{n}_{\mathbb{F},\tau}$ of dimension $\dim\mathbb{F}-1$ are congruent in $\mathsf{S}^{n}_{\mathbb{F},\tau}$ by some element in the isotropy of $\mathsf{S}^{n}_{\mathbb{F},\tau}$.

Proof.

Let us regard, up to a homothetical factor, the Hopf-Berger sphere $\mathsf{S}^{n}_{\mathbb{F},\tau}$ as a geodesic sphere $\mathsf{S}_{t}(p)$ of a rank one symmetric space $\bar{M}$ with maximal sectional curvature equal to $4$. Now taking into account that $\mathsf{S}^{n}_{\mathbb{F},\tau}$ and $\mathsf{S}_{t}(p)$ are related by a homothety of ratio $\alpha=\sqrt{1-\tau}$, see Equation (7); we deduce that $\widehat{\Sigma}$ (as constructed above) is isometric to $\mathsf{S}^{\dim\mathbb{F}-1}_{4-4\tau}$.

Let $v\in T_{o}\Sigma$ be a unit vector. Then, there exists a unique Helgason sphere $\mathsf{S}^{\dim\mathbb{F}}_{4}$ of $\bar{M}$ with $v\in T_{o}\mathsf{S}_{4}^{\dim\mathbb{F}}$. We may asumme that $v\in T_{o}\widehat{\Sigma}$ by using the isotropy of $\mathsf{S}^{n}_{\mathbb{F},\tau}$, and therefore $\widehat{\Sigma}\subset\mathsf{S}_{4}^{\dim\mathbb{F}}$. If $\Sigma$ is not contained in $\widehat{\Sigma}$, we have that $\dim(T_{o}\Sigma+T_{o}\widehat{\Sigma})\geq\dim\mathbb{F}$. Moreover, this implies that $\dim((T_{o}\Sigma\ominus\mathbb{R}v)+(T_{o}\widehat{\Sigma}\ominus\mathbb{R}v))\geq\dim\mathbb{F}-1$. Since $(T_{o}\Sigma\ominus\mathbb{R}v)+(T_{o}\widehat{\Sigma}\ominus\mathbb{R}v)$ is contained in the eigenspace associated with the eigenvalue $4$ of the Jacobi operator $\bar{R}_{v}$ (see Subsection 2.3), we deduce that $\dim(T_{o}\Sigma+T_{o}\widehat{\Sigma})=\dim\mathbb{F}$, and $T_{o}\Sigma+T_{o}\widehat{\Sigma}$ must be equal to the tangent space of $\mathsf{S}_{4}^{\dim\mathbb{F}}$. However, this yields a contradiction since this Helgason sphere $\mathsf{S}_{4}^{\dim\mathbb{F}}$ intersects the geodesic sphere $\mathsf{S}_{t}(p)$ transversally at some point, as we pointed out in the paragraph above Equation (22). This proves i).

Finally, notice that i) implies ii) since the isotropy at $o$ of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ acts transitively on the set of vectors with the same length and slope.∎

6. Totally geodesic submanifolds in quaternionic Hopf-Berger spheres

The purpose of this section is to achieve the classification of totally geodesic submanifolds in $\mathsf{S}^{4n+3}_{\mathbb{H},\tau}$.

6.1. Not well-positioned totally geodesic projective spaces

In what follows, we classify not well-positioned totally geodesic submanifolds in $\mathsf{S}^{4n+3}_{\mathbb{H},\tau}$ isometric to projective spaces.

First of all, notice that a not well-positioned totally geodesic projective space $\Sigma$ in $\mathsf{S}^{4n+3}_{\mathbb{H},\tau}$ has constant sectional curvature equal to $(1-\tau)$, since not well-positioned totally geodesic spheres have sectional curvature equal to $4(1-\tau)$, see Theorem 5.7 and Theorem 5.10. Let $\mathfrak{p}_{\Sigma}\subset\mathfrak{p}$ be the tangent space of $\Sigma$ at $o$. Then, we can define

[TABLE]

Since $\Sigma$ has constant sectional curvature equal to $(1-\tau)$, we have $T_{X}(Y)=0$ for every $X,Y\in\mathfrak{p}_{\Sigma}$.

As a consequence of the classification of totally geodesic submanifolds in rank one symmetric space (see Figure 4), Corollary 4.5, and Theorem 5.7, every not well-positioned totally geodesic projective space of $\mathsf{S}^{4n+3}_{\mathbb{H},\tau}$ has dimension two or three. Let us first start with the $2$-dimensional case, i.e. projective planes.

Lemma 6.1**.**

A not well-positioned totally geodesic surface of $\mathsf{S}^{4n+3}_{\mathbb{H},\tau}$ passing through $o\in\mathsf{S}_{\mathbb{H},\tau}^{4n+3}$ is isometric to a projective plane $\mathbb{R}\mathsf{P}^{2}_{1-\tau}$ if and only if it is congruent to $\Sigma=\exp_{o}(\mathfrak{p}_{\Sigma})$ by some element of the full isotropy of $\mathsf{S}^{4n+3}_{\mathbb{H},\tau}$, where $\mathfrak{p}_{\Sigma}$ is spanned by one of the following orthogonal bases:

[TABLE]

where $c_{m}:=1-m\tau$ for each $m\in\{1,2,3\}$.

Proof.

Observe that the subspaces spanned by $\mathcal{A}^{2}_{\tau}$, $\tau<1/3$ and $\mathcal{A}^{2}_{1/3}$ are $\upalpha$-isotropic, curvature invariant, and their sectional curvatures are equal to $(1-\tau)$, and $2/3$, respectively. Then, by Theorem 5.7, the exponential image of both subspaces are totally geodesic projective planes.

Conversely, notice that using the isotropy representation and the fact that $\Sigma$ is not well-positioned, we can assume without loss of generality that $\mathfrak{p}_{\Sigma}$ is spanned by two vectors $u$ and $v$ given by

[TABLE]

Recall that $\Sigma$ is isometric to a real projective plane, and therefore $\Sigma$ is locally symmetric. Thus,

[TABLE]

Taking into account that $\Sigma$ is not well-positioned, we have that $\tau<1/2$ by Theorem 5.7, and $a_{4}\neq 0$ by Lemma 4.1. Thus, $a_{1}=0$ or $b_{4}=0$. Let us assume that $b_{4}=0$. Then, we have for each $i\in\{1,2,3\}$,

[TABLE]

This implies the existence of a horizontal vector that yields a contradiction by Lemma 4.1. Let us assume that $a_{1}=0$. Then,

[TABLE]

Hence, $b_{1}=0$, since otherwise $\mathfrak{p}_{\Sigma}$ would contain a non-zero horizontal vector. Moreover,

[TABLE]

This implies that $a_{2}=r\cos(\varphi)$, $a_{3}=r\sin(\varphi)$, $b_{2}=-s\sin(\varphi)$, $b_{3}=s\cos(\varphi)$, for some $r,s>0$ and $\varphi\in[0,2\pi)$. However, we can assume without loss of generality, that $r=s=1$. By using the isotropy, we can further assume that $\varphi=0$. Hence,

[TABLE]

This implies that $a_{4}=\pm\frac{\sqrt{c_{2}}}{\sqrt{\tau}}$. Then, we have

[TABLE]

which implies that $b_{4}=\pm\frac{\sqrt{\tau}}{\sqrt{c_{2}}}$. By using the isotropy, we can assume that both $a_{4}$ and $b_{4}$ are positive. Finally, since $\mathfrak{p}_{\Sigma}$ is $\upalpha$-isotropic, we have

[TABLE]

This implies that $b_{5}=\pm\sqrt{\frac{c_{1}c_{3}}{\tau c_{2}}}$. Observe that $b_{5}\in\mathbb{R}$ when $\tau<1/3$. Using the isotropy we can assume that $b_{5}>0$, and then we obtain the subspace $\mathfrak{p}_{\Sigma}$ spanned by $\mathcal{A}^{2}_{\tau}$. On the one hand, observe that $b_{5}=0$ when $\tau=1/3$, obtaining the subspace spanned by $\mathcal{A}^{2}_{1/3}$. On the other hand, if $\tau<1/3$, we have $b_{5}\neq 0$, and then $\Sigma$ is a totally geodesic submanifold of $\mathsf{S}^{11}_{\mathbb{H},\tau}$. This proves the desired result. ∎

Lemma 6.2**.**

A not well-positioned totally geodesic submanifold of $\mathsf{S}^{4n+3}_{\mathbb{H},\tau}$ passing through $o\in\mathsf{S}^{4n+3}_{\mathbb{H},\tau}$ is isometric to a projective space $\mathbb{R}\mathsf{P}^{3}_{1-\tau}$ if and only if it is congruent to $\Sigma=\exp_{o}(\mathfrak{p}_{\Sigma})$ by some element of the full isotropy of $\mathsf{S}^{4n+3}_{\mathbb{H},\tau}$, where $\mathfrak{p}_{\Sigma}$ is spanned by one of the following orthogonal bases:

[TABLE]

where $c_{m}:=1-m\tau$ for each $m\in\{1,2,3,4\}$.

Proof.

Observe that the subspaces spanned by $\mathcal{A}^{3}_{\tau}$, $\tau<1/4$, and $\mathcal{A}^{2}_{1/4}$ are $\upalpha$-isotropic, curvature invariant, and their sectional curvatures are equal to $(1-\tau)$ and $3/4$, respectively. Then, by Theorem 5.7, the exponential image of both subspaces are three dimensional totally geodesic projective spaces.

Conversely, using the isotropy, the fact that $\Sigma$ is not well-positioned and Lemma 6.1, we can assume without loss of generality that $\mathfrak{p}_{\Sigma}$ is spanned by three vectors $u$, $v$ and $w$ given by

[TABLE]

Recall that $\Sigma$ has constant sectional curvature and therefore $\Sigma$ is locally symmetric. Thus,

[TABLE]

This implies that $a_{4}=a_{6}=0$. Now assume that $\tau=1/3$. Then,

[TABLE]

Thus, $a_{5}=0$. However,

[TABLE]

yielding a contradiction with the fact that $\tau=1/3$. Hence, we can assume that $\tau\neq 1/3$. Then,

[TABLE]

Consequently, $a_{11}=0$. Moreover,

[TABLE]

which implies that $a_{5}=0$ and $a_{7}=-\sqrt{\tau}/\sqrt{c_{2}}$. Now,

[TABLE]

Therefore, we deduce that $a_{8}=a_{9}=0$. Furthermore,

[TABLE]

Hence, we deduce that $a_{10}=\frac{\sqrt{c_{1}\tau}}{\sqrt{c_{2}c_{3}}}$. Finally, taking into account that $\mathfrak{p}_{\Sigma}$ is $\upalpha$-isotropic, we have

[TABLE]

Consequently, we have $a_{12}=\pm\sqrt{\frac{c_{1}c_{4}}{\tau c_{3}}}$. Using the isotropy we can assume that $a_{12}>0$, and then we obtain the subspace $\mathfrak{p}_{\Sigma}$ spanned by $\mathcal{A}^{3}_{\tau}$. On the one hand, observe that $a_{12}=0$ when $\tau=1/4$, obtaining the subspace spanned by $\mathcal{A}^{3}_{1/4}$. On the other hand, if $\tau<1/4$, we have $a_{12}\neq 0$, and then $\Sigma$ is a totally geodesic submanifold of $\mathsf{S}^{15}_{\mathbb{H},\tau}$. This proves the desired result.∎

Proposition 6.3**.**

Every two isometric totally geodesic submanifolds of $\mathsf{S}^{4n+3}_{\mathbb{H},\tau}$ passing by $o$ with dimension $d\geq 2$ are isotropy-congruent at $o$.

Proof.

Firstly, the result follows for well-positioned totally geodesic submanifolds by virtue of Remark 4.7. By Lemma 6.1 and Lemma 6.2, two isometric not well-positioned totally geodesic submanifolds of $\mathsf{S}^{4n+3}_{\mathbb{H},\tau}$ diffeomorphic to projective spaces passing through $o$ are isotropy-congruent at $o$. By Theorem 5.10, there is exactly one isotropy-congruence class of not well-positioned totally geodesic $3$-dimensional spheres.

Finally, we tackle the case of $2$-dimensional not well-positioned spheres. Firstly, we claim that not well-positioned $3$-dimensional totally geodesic spheres are orbits of a subgroup of $\mathsf{S}^{4n+3}_{\mathbb{H},\tau}$ isomorphic to $\mathsf{SO}_{4}$. By Lemma 3.4 and Theorem 5.10, every not well-positioned totally geodesic $3$-dimensional sphere $\widehat{\Sigma}$ is congruent to the orbit of some subgroup $\mathsf{H}$ of $\operatorname{Isom}(\mathsf{S}^{4n+3}_{\mathbb{H},\tau})$ by $o\in\mathsf{S}^{4n+3}_{\mathbb{H},\tau}$. By Remark 5.9, we can assume without loss of generality that $n=1$. Now take some non-zero element $X$ in $T_{o}\widehat{\Sigma}$. Let $\mathsf{H}_{o}$ be the isotropy of $\mathsf{H}$ at $o$. Let $X,Y$ in $T_{o}\widehat{\Sigma}$ be of unit length. Since $X$ and $Y$ have the same slope, see Lemma 5.4, we have an element $k$ of the isotropy of $\mathsf{S}^{n}_{\mathbb{F},\tau}$ such that $k_{*o}X=Y$. By the uniqueness of the totally geodesic submanifold $\widehat{\Sigma}$ (see Theorem 5.10), we have that $k\widehat{\Sigma}=\widehat{\Sigma}$. This proves that $\widehat{\Sigma}$ is extrinsically $2$-point homogeneous. Therefore, by the classification of transitive actions on spheres, see [38], $\mathsf{H}$ is locally isomorphic to $\mathsf{SO}_{4}$.

Now, observe that $\mathsf{H}$ must act effectively on $\widehat{\Sigma}$, otherwise there would be some $\varphi\in\mathsf{H}$ such that $\widehat{\Sigma}\subset\mathrm{Fix}(\varphi)$, and the set of fixed points of an isometry is a well-positioned totally geodesic submanifold, see Lemma 4.8. Thus, $\mathsf{H}$ must be isomorphic to $\mathsf{SO}_{4}$.

By Theorem 5.10, every not well-positioned two-dimensional sphere can be contained in $\widehat{\Sigma}$. Now since $\mathsf{H}_{o}=\mathsf{SO}_{3}$, there is some element $\varphi\in\mathsf{H}_{o}$ acting transitively on the $2$-plane Grassmannian $\mathsf{G}_{2}(T_{o}\widehat{\Sigma})\equiv\mathsf{G}_{2}(\mathbb{R}^{3})$, and therefore there is just one isotropy-congruence class at $o$ for not well-positioned totally geodesic spheres of dimension two passing through the point $o$.∎

Since Hopf-Berger spheres are g.o. spaces, as a consequence of Proposition 6.3 and Lemma 3.4, we have the following.

Corollary 6.4**.**

Every totally geodesic submanifold of $\mathsf{S}^{4n+3}_{\mathbb{H},\tau}$ is extrinsically homogeneous.

We end up this section with the proof of Theorem B.

Proof of Theorem B.

On the one hand, if $\Sigma$ is well-positioned the proof follows by Theorem 4.4, Remark 4.6, and Remark 4.7. On the other hand, if $\Sigma$ is not well-positioned the proof follows by combining Theorem 5.7, Lemma 6.1, Lemma 6.2 and Proposition 6.3. ∎

7. Totally geodesic submanifolds in octonionic Hopf-Berger spheres

The purpose of this section is to achieve the classification of totally geodesic submanifolds in $\mathsf{S}^{15}_{\mathbb{O},\tau}$.

Firstly, let us prove that the totally geodesic submanifold $\widehat{\Sigma}$ of $\mathsf{S}^{15}_{\mathbb{O},\tau}$ (see Theorem 5.10) is an orbit of a subgroup $\mathsf{H}$ of $\mathsf{Spin}_{9}$ isomorphic to $\mathsf{Spin}_{7}$. By Lemma 3.4 and Theorem 5.10, $\widehat{\Sigma}$ is extrinsically homogeneous. According to [47], if $\tau\neq 1$, we have the following situation for certain subgroups of $\operatorname{Isom}(\mathsf{S}^{15}_{\mathbb{O},\tau})=\mathsf{Spin}_{9}$,

[TABLE]

In this commuting diagram the lower left corner of each square is the intersection of the subgroup that is above and the subgroup to the right of it. Every inclusion in this diagram is maximal. Moreover, it is well-known that apart from the natural inclusion $i$ of $\mathsf{Spin}_{7}$ in $\mathsf{Spin}_{8}$, there are $2$ more non-standard inclusions $i^{\prime}$ and $i^{\prime\prime}$, which together with $i$ lead to the $3$ distinct conjugacy classes of $\mathsf{Spin}_{7}$-subgroups in $\mathsf{Spin}_{8}$. These are related by the triality group $\mathrm{Out}(\mathsf{Spin}_{8})$ which acts transitively on the $3$-conjugacy classes of its $\mathsf{Spin}_{7}$-subgroups, see [47].

Let $\mathsf{K}_{v}$ be the stabilizer for the isotropy action of $\mathsf{S}^{15}_{\mathbb{O},\tau}=\mathsf{G}/\mathsf{K}$ at a unit $\upalpha$-isotropic vector $v\in T_{o}\widehat{\Sigma}$. The stabilizer for the isotropy action of $\mathsf{S}^{15}_{\mathbb{O},\tau}$ at a non-zero vertical vector or at a horizontal vector is isomorphic to $\mathsf{Spin}_{6}$ or $\mathsf{G}_{2}$, respectively. Then $\mathsf{K}_{v}$ is isomorphic to the intersection of $\mathsf{Spin}_{6}$ and $\mathsf{G}_{2}$, which is $\mathsf{SU}_{3}$ by the previous discussion. Recall that $\mathsf{H}$ is the subgroup of $\mathsf{Spin}_{9}$ such that $\mathsf{H}\cdot o=\widehat{\Sigma}$. By a similar argument as in Proposition 6.3, using the uniqueness of $\widehat{\Sigma}$ (see Theorem 5.10), we have that $\widehat{\Sigma}$ is an extrinsically $2$-point homogeneous submanifold of $\mathsf{S}^{15}_{\mathbb{O},\tau}$. Then, $\mathsf{H}{{}_{o}}v=\mathsf{\widetilde{H}}{{}_{o}}/\mathsf{\widetilde{K}}_{v}=\mathsf{S}(T_{o}\widehat{\Sigma})=\mathsf{S}^{6}$, where $\widetilde{\mathsf{H}}_{o}$ and $\mathsf{\widetilde{K}}_{v}$ denote the effectivized connected subgroups corresponding to the action of $\mathsf{H}_{o}$ and $\mathsf{K}_{v}$ on $T_{o}\widehat{\Sigma}$, respectively. Since $\mathsf{K}_{v}$ is simple, then $\widetilde{\mathsf{K}}_{v}$ is either trivial or locally isomorphic to $\mathsf{SU}_{3}$. In the first case, we would obtain a contradiction since $\mathsf{S}^{6}$ is not a Lie group. Thus, $\widetilde{\mathsf{K}}_{v}$ is locally isomorphic to $\mathsf{SU}_{3}$. Consequently, by the classification of transitive actions on spheres, see [38], $\widetilde{\mathsf{H}}_{o}$ is isomorphic to $\mathsf{G}_{2}$ and $\widetilde{\mathsf{K}}_{v}$ is isomorphic to $\mathsf{SU}_{3}$. Now, observe that $\mathsf{H}$ must act effectively on $\widehat{\Sigma}$, otherwise there would be some $\varphi\in\mathsf{H}$ such that $\widehat{\Sigma}\subset\mathrm{Fix}(\varphi)$. The set of fixed points of an isometry is a well-positioned totally geodesic submanifold which has dimension less or equal than $7$, see Lemma 4.8 and Table 1. Thus, $\mathsf{H}$ must be isomorphic to $\mathsf{Spin}_{7}$ by the classification of transitive actions on spheres.

The following lemma is a crucial step towards the classification of totally geodesic submanifolds in $\mathsf{S}^{15}_{\mathbb{O},\tau}$.

Lemma 7.1**.**

Let $\widehat{\Sigma}$ be a not well-positioned totally geodesic sphere in $\mathsf{S}^{15}_{\mathbb{F},\tau}$ of dimension $7$. Then, the following statements hold:

1. i)

The totally geodesic submanifold $\widehat{\Sigma}$ is an orbit of a subgroup of $\mathsf{Spin}_{9}$ isomorphic to $\mathsf{Spin}_{7}$ and the isotropy of the action of $\mathsf{Spin}_{7}$ on $\mathsf{S}^{15}_{\mathbb{O},\tau}$ at $o\in\widehat{\Sigma}$ is isomorphic to $\mathsf{G}_{2}$. 2. ii)

Let $\Sigma$ be a totally geodesic submanifold of $\mathsf{S}^{15}_{\mathbb{O},\tau}$ with dimension greater than two. Then, $\Sigma$ is not well-positioned if and only if it is congruent to a totally geodesic submanifold of $\widehat{\Sigma}$. 3. iii)

The trilinear map $\upvarphi$ defined by

[TABLE]

is a $\mathsf{G}_{2}$-invariant $3$-form.

Proof.

The statement in i) is proved in the discussion prior to this lemma.

Clearly, every totally geodesic submanifold $\Sigma$ of $\widehat{\Sigma}$ is not well-positioned. Conversely, let $\Sigma$ be a not well-positioned of dimension greater or equal than three. By Theorem 5.7, $\Sigma$ contains a totally geodesic surface $N$. By Lemma 4.10, the tangent space of $N$ is equal to the tangent space of a not well-positioned totally geodesic surface of $\mathsf{S}^{7}_{\mathbb{H},\tau}$. Therefore, $T_{o}N$ is either equal to a 2-dimensional subspace of the tangent space of a totally geodesic submanifold congruent to $\widehat{\Sigma}$ by Theorem 5.10, or $\tau=1/3$ and $T_{o}N$ is equal to the tangent space of a not well-positioned totally geodesic projective space in a well-positioned totally geodesic $\mathsf{S}^{7}_{\mathbb{H},1/3}$ as in Lemma 6.1 (b).

Thus, assume that the tangent space at $o\in\mathsf{S}^{15}_{\mathbb{O},\tau}$ of $N$ is as in Lemma 6.1 (b). Then, $\tau=1/3$ and $T_{o}N$ is spanned by $u=\frac{\sqrt{3}}{2}X_{2}+Y_{1}$ and $v=\frac{\sqrt{3}}{2}X_{3}+Y_{5}$. Let $w\in\mathfrak{p}_{\Sigma}$ be a non-zero vector orthogonal to $u$ and $v$. Let us write $w=\sum_{i=1}^{7}\alpha_{i}\frac{\sqrt{3}}{2}X_{i}+\sum_{i=1}^{8}\beta_{i}Y_{i}$. Now using that $\Sigma$ is a symmetric space,

[TABLE]

Thus,

[TABLE]

By Lemma 6.1, $N$ is isometric to $\mathbb{R}\mathsf{P}^{2}_{2/3}$. Moreover, $\Sigma$ contains $N$ as a totally geodesic submanifold, and thus, by Theorem 5.7 and Corollary 4.5 , $\Sigma$ is homothetic to $\mathbb{C}\mathsf{P}^{2}$. If $w\in\mathfrak{p}_{\Sigma}$ is a non-zero vector orthogonal to $X$ and $v$, we have $\sec(u,w)=\sec(v,w)=4\sec(u,v)$. Therefore,

[TABLE]

[TABLE]

where $\omega_{i}:=2\alpha^{2}_{4-i}+\alpha^{2}_{4}+\alpha^{2}_{5}+\alpha^{2}_{6}+\alpha^{2}_{7}+\beta_{2}^{2}+10\beta^{2}_{3}+\beta^{2}_{4}+\beta^{2}_{6}+\beta^{2}_{8}$, for each $i\in\{1,2\}$. Now adding Equation (23) and Equation (24), we get

[TABLE]

This implies that $w$ is zero yielding a contradiction. Thus, we proved ii).

Let $\upvarphi$ be defined as in iii). Then, the trilinear map $\upvarphi$ is a $3$-form, since we have the equalities $\upvarphi(X,X,Y)=\upvarphi(X,Y,X)=\upvarphi(Y,X,X)=0$ for every $X,Y\in T_{o}\widehat{\Sigma}$. Finally, notice that the isotropy of a not well-positioned totally geodesic submanifold of dimension $7$ is isomorphic to $\mathsf{G}_{2}$. The action of $\mathsf{G}_{2}$ preserves $\mathfrak{p}_{1}$ and $\mathfrak{p}_{2}$, since $\mathsf{G}_{2}$ lies in the isotropy of $\mathsf{S}^{15}_{\mathbb{O},\tau}$. Moreover, $\mathsf{G}_{2}$ preserves the metric and the Lie bracket of the Lie algebra $\mathfrak{spin}_{9}$. Consequently, $\upvarphi$ is a $\mathsf{G}_{2}$-invariant $3$-form. ∎

In the following we will study the congruence problem for totally geodesic submanifolds in $\mathsf{S}^{15}_{\mathbb{O},\tau}$.

Theorem 7.2**.**

The moduli space of all not well-positioned totally geodesic spheres in $\mathsf{S}^{15}_{\mathbb{O},\tau}$ of dimension greater or equal than two (up to congruence in $\mathsf{S}^{15}_{\mathbb{O},\tau}$) is isomorphic to the disjoint union

[TABLE]

The number $k\in\{2,4,5,6,7\}$ parametrizes the unique (up to congruence in $\mathsf{S}^{15}_{\mathbb{O},\tau}$) totally geodesic submanifold in $\mathsf{S}^{15}_{\mathbb{O},\tau}$ of dimension $k$. The set $\{3\}\times[0,1]$ parametrizes the not well-positioned totally geodesic submanifolds of dimension $3$ (up to congruence in $\mathsf{S}^{15}_{\mathbb{O},\tau}$).

Proof.

Let $\Sigma_{1}$ and $\Sigma_{2}$ be two not well-positioned totally geodesic spheres of $\mathsf{S}^{15}_{\mathbb{O},\tau}$ with dimension $k$. When $k=7$, $\Sigma_{1}$ and $\Sigma_{2}$ are congruent by Lemma 7.1 ii).

Now assume that $\Sigma_{1}$ and $\Sigma_{2}$ are two not well-positioned totally geodesic spheres of dimension $2$. By Lemma 4.10 and Lemma 7.1, they are contained in a well-positioned totally geodesic $\mathsf{S}^{7}_{\mathbb{H},\tau}$. Hence, $\Sigma_{1}$ and $\Sigma_{2}$ are congruent in $\mathsf{S}^{15}_{\mathbb{O},\tau}$ by Proposition 6.3.

In what follows, let $\Sigma_{1}$ and $\Sigma_{2}$ be two not well-positioned totally geodesic spheres of dimension $k\in\{3,4,5,6\}$. Then, by Theorem 5.10 and Lemma 7.1, the problem is reduced to analyze the congruence classes of $k$-dimensional totally geodesic submanifolds of the round sphere $\mathsf{S}^{7}$ under the subgroup $\mathsf{Spin}_{7}$. Observe that a $k$-dimensional totally geodesic submanifold of $\mathsf{S}^{7}$ is uniquely determined by a $(k+1)$-plane of $\mathbb{R}^{8}$. Moreover, $\mathsf{Spin}_{7}$ acts transitively on $\mathsf{G}_{k+1}(\mathbb{R}^{8})$, the Grassmannian of $(k+1)$-planes of $\mathbb{R}^{8}$, if and only if $k+1\neq 4$; and with cohomogeneity one when $k+1=4$. Furthermore this cohomogeneity one action has two singular orbits. Consequently, the set of congruence classes for $3$-dimensional not well-positioned totally geodesic spheres is identified with $[0,1]$ and for $k\neq 3$ there is just one congruence class. ∎

In the sequel, we explain how the $3$-form constructed in Lemma 7.1 iv) defines an invariant which allows to distinguish between non-congruent not well-positioned totally geodesic $3$-spheres of $\mathsf{S}^{15}_{\mathbb{O},\tau}$. Let us identify $\operatorname{Im}\mathbb{O}$ with $\mathbb{R}^{7}$ and consider the 3-form $\upvarphi$ given by

[TABLE]

where $\langle\cdot,\cdot\rangle$ denotes the Euclidean inner product in $\mathbb{R}^{7}$, and $\cdot$ denotes the product in $\mathbb{O}$. Also, $\upvarphi$ satisfies $||\upvarphi||\leq 1$, where ||\upvarphi||:=\sup\{\upvarphi(\xi):\text{\xi$is a unit, simple 3-vector in$\mathbb{R}^{7}}\}. This is a $\mathsf{G}_{2}$-invariant 3-form and it was called by Harvey and Lawson [20] the associative calibration on $\operatorname{Im}\mathbb{O}$. The reason for this name is the following. Let $\xi=x\wedge y\wedge z$, where $x,y$ and $z$ are elements of $\operatorname{Im}\mathbb{O}$ of unit length. Then, $|\upvarphi(\xi)|=1$ if and only if the $3$-dimensional subspace spanned by $\{x,y,z\}$ is the imaginary part of a subalgebra of $\mathbb{O}$ isomorphic to $\mathbb{H}$.

We can consider the smooth function $\upvarphi\colon\mathsf{G}_{3}\mathbb{R}^{7}\rightarrow[-1,1]$, where $\mathsf{G}_{3}\mathbb{R}^{7}$ denotes the $3$-plane Grassmannian over $\mathbb{R}^{7}$. A direct computation shows that the critical points of $\upvarphi$ are precisely the associative $3$-planes. Moreover, $\mathsf{G}_{2}$ acts on $\mathsf{G}_{3}\mathbb{R}^{7}$ with cohomogeneity one (see [32, Table 1]). Thus, $|\upvarphi(V)|=|\upvarphi(V^{\prime})|$ if and only if $V$ and $V^{\prime}$ in $\mathsf{G}_{3}\mathbb{R}^{7}$ belong to the same orbit for the $\mathsf{G}_{2}$ action on $\mathsf{G}_{3}\mathbb{R}^{7}$.

Furthermore, notice that the representation of $\mathsf{G}_{2}$ on $\Lambda^{3}(\mathbb{R}^{7})$ splits as the direct sum of the trivial, $7$-dimensional, and $27$-dimensional irreducible representations of $\mathsf{G}_{2}$, see [43, Theorem 8.5]. Thus, there is just one (up to scaling) $\mathsf{G}_{2}$-invariant $3$-form defined in $\operatorname{Im}\mathbb{O}$. Moreover, the $3$-form $\upvarphi$ defined in Lemma 7.1 iv) also satisfies that $||\upvarphi||\leq 1$, so it agrees with the associative calibration up to sign. Consequently, given two not well-positioned totally geodesic $3$-spheres $\Sigma_{1}=\exp(V_{1})$ and $\Sigma_{2}=\exp(V_{2})$ contained in the same maximal not well-positioned totally geodesic sphere of $\mathsf{S}^{15}_{\mathbb{O},\tau}$, we have that they are congruent if and only if $|\upvarphi(V_{1})|=|{\upvarphi}(V_{2})|$. Furthermore, we can always find an element $g\in\mathsf{Spin}_{7}$ such that $\Sigma_{1}$ and $g\Sigma_{2}$ are contained in the same maximal not well-positioned totally geodesic sphere, and since $\upvarphi$ is also invariant under $\mathsf{Spin}_{7}$, we have the following result.

Proposition 7.3**.**

Let $\Sigma_{i}=\exp(V_{i})$ be a not well-positioned totally geodesic $3$-sphere of $\mathsf{S}^{15}_{\mathbb{O},\tau}$ for each $i\in\{1,2\}$. Then, $\Sigma_{1}$ and $\Sigma_{2}$ are congruent in $\mathsf{S}^{15}_{\mathbb{O},\tau}$ if and only if $|\upvarphi(V_{1})|=|\upvarphi(V_{2})|$.

We ilustrate the usefulness of the invariant defined above with the following example.

Example 7.4*.*

Let us consider the basis $\{v_{i}\}^{7}_{i=1}$ of $T_{o}\widehat{\Sigma}$ as defined by Equation (22). Now for $\theta\in[0,1]$ consider the subspaces $V_{\theta}$ and $V^{\prime}_{\theta}$ spanned by

[TABLE]

respectively. It turns out that $\upvarphi(V_{\theta})=\sin(\tfrac{\pi\theta}{2})$ and $\upvarphi(V^{\prime}_{\theta})=\cos(\tfrac{\pi\theta}{2})\in[0,1]$. Therefore, $\Sigma_{\theta}=\exp_{o}(V_{\theta})$ and $\Sigma_{\theta}^{\prime}=\exp_{o}(V^{\prime}_{\theta})$ are congruent if and only if $\theta=1/2$.

Remark 7.5*.*

We observe the following interesting fact. Let $V$ be a $3$-dimensional subspace of $T_{o}\widehat{\Sigma}$. Then, $\Sigma=\exp_{o}(V)$ is a not well-positioned totally geodesic submanifold $3$-sphere of a $7$-dimensional well-positioned totally geodesic submanifold isometric to $\mathsf{S}^{7}_{\mathbb{H},\tau}$ of $\mathsf{S}^{15}_{\mathbb{O},\tau}$ if and only if $\upvarphi(V)=1$, i.e. $V$ is associative when regarded as a subspace of $\operatorname{Im}\mathbb{O}$.

This follows by considering the subspace $V_{\theta}$ with $\theta=1$ in Example 7.4 and the subspace given by $(1)_{\mathbb{O}}$ in Subsection 4.1 combined with the uniqueness (up to congruence) of the totally geodesic inclusions $\mathsf{S}^{7}_{\mathbb{H},\tau}\hookrightarrow\mathsf{S}^{15}_{\mathbb{O},\tau}$ and $\mathsf{S}^{3}_{4-4\tau}\hookrightarrow\mathsf{S}^{7}_{\mathbb{H},\tau}$, respectively; see Remark 4.7 and Proposition 6.3.

Now we are in a position to prove Theorem C.

Proof of Theorem C.

On the one hand, if $\Sigma$ is well-positioned the proof follows by Theorem 4.4, Remark 4.6, and Remark 4.7. On the other hand, if $\Sigma$ is not well-positioned the proof follows by combining Theorem 5.7, Lemma 7.1 and Theorem 7.2. ∎

To conclude this section we will discuss the homogeneity of the not well-positioned totally geodesic submanifolds of $\mathsf{S}^{15}_{\mathbb{O},\tau}$.

Theorem 7.6**.**

Let $\Sigma$ be a totally geodesic submanifold of $\mathsf{S}^{15}_{\mathbb{O},\tau}$. Then, $\Sigma$ is not extrinsically homogeneous if and only if $\Sigma$ is congruent to a $4$-dimensional not well-positioned totally geodesic sphere.

Proof.

Firstly, let us assume that $\Sigma$ is equal to a $4$-dimensional not well-positioned totally geodesic sphere of $\mathsf{S}^{15}_{\mathbb{O},\tau}$. By Theorem 5.10, we can always assume that $\Sigma$ is contained in a $7$-dimensional not well-positioned totally geodesic sphere $\mathsf{S}^{7}_{4-4\tau}$ of $\mathsf{S}^{15}_{\mathbb{O},\tau}$. Then, by Lemma 7.1, we can reduce the discussion to decide if a totally geodesic $\mathsf{S}^{4}$ in a round $\mathsf{S}^{7}=\mathsf{Spin}_{7}/\mathsf{G}_{2}$ is an orbit of some subgroup of $\mathsf{Spin}_{7}$.

Let $\mathsf{K}=\{g\in\mathsf{SO}_{8}:g\mathsf{S}^{4}=\mathsf{S}^{4}\}^{0}$, where $\cdot^{0}$ denotes the connected component containing the identity. Then $\mathsf{K}=\mathsf{K}_{1}\times\mathsf{K}_{2}$ where $\mathsf{K}_{1}\simeq\mathsf{SO}_{5}$ and $\mathsf{K}_{2}\simeq\mathsf{SO}_{3}$. Notice that $\mathsf{K}_{1}$ acts on $\mathsf{S}^{4}\subset\mathbb{R}^{5}$, and $\mathsf{K}_{2}$ acts on $\mathbb{R}^{3}\simeq\mathbb{R}^{8}\ominus\mathbb{R}^{5}$. Let us assume that there exists a Lie subgroup $\mathsf{F}$ of $\mathsf{Spin}_{7}\subset\mathsf{SO}_{8}$ that acts transitively on $\mathsf{S}^{4}$ and let $\operatorname{proj}_{1}\colon\mathsf{F}\to\mathsf{K}_{1}$ be the projection. Let $\operatorname{proj}_{1}\colon\mathfrak{f}\rightarrow\mathfrak{k}_{1}$ be the corresponding map at the Lie algebra level. Since $\mathsf{F}$ is compact, we may assume that $\mathsf{F}$ acts almost effectively on $\mathsf{S}^{4}$. Indeed, if the action of $\mathsf{F}$ is not almost effectively on $\mathsf{S}^{4}$, we can replace $\mathsf{F}$ by the Lie group associated with a complementary ideal in $\mathfrak{f}$ of $\mathfrak{f}\cap\mathfrak{k}_{2}=\ker\operatorname{proj}_{1}$. Then $\operatorname{proj}_{1}:\mathsf{F}\to\mathsf{K}_{1}$ is almost injective and $\operatorname{proj}_{1}(\mathsf{F})=\mathsf{SO}_{5}$, since $\mathsf{SO}_{5}$ is the only compact Lie group acting transitively and effectively on $\mathsf{S}^{4}$ by the classification of transitive actions on spheres, see [38]. Then $\mathfrak{f}$ is isomorphic to $\mathfrak{so}_{5}$. Hence, since $\mathfrak{f}$ is simple and $\dim(\mathfrak{f})>\dim(\mathfrak{so}_{3})$ the Lie algebra morphism $\operatorname{proj}_{2}:\mathfrak{f}\to\mathfrak{k}_{2}$ is trivial. This proves that $\mathfrak{f}=\mathfrak{k}_{1}$. Let us consider the Cartan decomposition of the symmetric spaces $\mathsf{S}^{4}$ and $\mathsf{S}^{7}$, given by $\mathfrak{k}_{1}=\mathfrak{p}_{1}\oplus\mathfrak{h}_{1}$ and $\mathfrak{so}_{8}=\mathfrak{p}\oplus\mathfrak{so}_{7}$, respectively. Let us choose a non-zero element $X\in\mathfrak{p}_{1}$. Then, $X\in\mathfrak{p}\cap\mathfrak{spin}_{7}$. Thus, since $\mathsf{G}_{2}$ acts transitively on the unit sphere of $T_{o}\mathsf{S}^{7}$, for each direction $X\in T_{o}\mathsf{S}^{7}$, there exists a Killing vector field of $\mathsf{S}^{7}$ parallel at $o$. Consequently, $\mathfrak{f}$ contains $\mathfrak{p}$, and thus $\mathfrak{f}$ contains $\mathfrak{so}_{8}=\mathfrak{p}\oplus[\mathfrak{p},\mathfrak{p}]$ yielding a contradiction.

Secondly, we will prove that every totally geodesic submanifold of $\mathsf{S}^{15}_{\mathbb{O},\tau}$ (see Table 3) that is not congruent to a $4$-dimensional not well-positioned totally geodesic sphere is extrinsically homogeneous. The geodesics of $\mathsf{S}^{15}_{\mathbb{O},\tau}$ are orbits, since $\mathsf{S}^{15}_{\mathbb{O},\tau}$ is a g.o. space; and well-positioned totally geodesic submanifolds of $\mathsf{S}^{15}_{\mathbb{O},\tau}$ are also orbits, by Remark 4.7.

Let $\Sigma$ be congruent to the totally geodesic projective plane $\mathbb{R}\mathsf{P}^{2}_{2/3}$. Then $\tau=1/3$, see Lemma 6.1. In this case $\Sigma$ is contained in a well-positioned totally geodesic submanifold of $\mathsf{S}^{15}_{\mathbb{O},1/3}$ isometric to $\mathsf{S}^{7}_{\mathbb{H},1/3}$ by Lemma 4.10. Then, $\Sigma$ is an extrinsically homogeneous submanifold of $\mathsf{S}^{15}_{\mathbb{O},\tau}$.

Let $\Sigma$ be congruent to a not well-positioned totally geodesic sphere of $\mathsf{S}^{15}_{\mathbb{O},\tau}$. By Theorem 5.10, we can always assume that $\Sigma$ is contained in a $7$-dimensional not well-positioned totally geodesic sphere of $\mathsf{S}^{15}_{\mathbb{O},\tau}$. Then, Lemma 7.1 implies that we can reduce the discussion to analyze which totally geodesic submanifolds of the round sphere $\mathsf{S}^{7}$ are subgroups of $\mathsf{Spin}_{7}$. Let us assume that $\Sigma$ is isometric to a not well-positioned totally geodesic sphere. Let $k\neq 4$ be the dimension of $\Sigma$. Now, if $k=7$, $\Sigma$ is extrinsically homogeneous by Theorem 5.10 and Lemma 3.4.

Observe that for every $k$-dimensional subspace $V^{k}\subset T_{o}\mathsf{S}^{7}$ there is a totally geodesic $\mathsf{S}^{k}$ such that $T_{o}\mathsf{S}^{k}=V^{k}$. Let us consider the action of $\mathsf{G}_{2}$ on the Grassmanian $\mathsf{G}_{k}(T_{o}\mathsf{S}^{7})$ of $k$-subspaces of $T_{o}\mathsf{S}^{7}$. If $k\not\in\{3,4\}$, then this action is transitive (see [26, Lemma 1.1]). Hence, there is only one congruence class at $o$ of totally geodesic $\mathsf{S}^{k}$. Then, by Lemma 3.4, $\mathsf{S}^{k}$ is extrinsically homogeneous for $k\not\in\{3,4\}$.

Let us consider the action of $\mathsf{G}_{2}$ on the Grassmannian $\mathsf{G}_{3}(T_{o}\mathsf{S}^{7})$ of $3$-planes in $T_{o}\mathsf{S}^{7}$. It is known that this action has cohomogenity one (see e.g. [32, Table 1]). The isotropy subgroup $(\mathsf{G}_{2})_{\pi}$ at any $3$-plane $\pi$ has dimension at least $3$, since $\dim(\mathsf{G}_{2})=14$ and $\dim(\mathsf{G}_{3}(T_{o}\mathsf{S}^{7}))=12$. Moreover, since $(\mathsf{G}_{2})_{\pi}\subset\mathsf{G}_{2}$, the rank of $(\mathsf{G}_{2})_{\pi}$ is at most $2$ and thus $(\mathsf{G}_{2})_{\pi}$ is not abelian. Hence, $(\mathsf{G}_{2})_{\pi}$ has a connected and (non-trivial) compact simple normal subgroup $\mathsf{K}^{\pi}$. Note that the isotropy of a principal orbit $\mathsf{G}_{2}\cdot\pi$ has dimension $3$, and thus the Lie algebra of $\mathsf{K}^{\pi}$ is isomorphic to $\mathfrak{so}_{3}$.

Let $\pi$ be an arbitrary $3$-plane of $T_{o}\mathsf{S}^{7}$ and let $\mathsf{S}^{3}$ be the $3$-dimensional totally geodesic sphere of $\mathsf{S}^{7}$ with $T_{o}\mathsf{S}^{3}=\pi$. Let us consider the action of $\mathsf{K}^{\pi}$ on $T_{o}\mathsf{S}^{7}$. Observe that this action leaves invariant the subspace $\pi$. Now we will consider the case in which $\mathsf{K}^{\pi}$ acts trivially on $\pi$, and the case in which $\mathsf{K}^{\pi}$ acts non-trivially on $\pi$.

Case (i): $\mathsf{K}^{\pi}$ acts trivially on $\pi$. Let $V\subset T_{o}\mathsf{S}^{7}$ be the subspace of fixed vectors of $\mathsf{K}^{\pi}$. Then $V$ contains $\pi$. If ${V}=\pi$, then $\mathsf{S}^{3}$ is extrinsically homogeneous since $\mathsf{S}^{3}$ is a connected component of the fixed set of a subgroup of the presentation group of $\mathsf{S}^{7}$, in this case $\mathsf{Spin}_{7}$, see [8, Lemma 9.1.1]. Hence, if $V=\pi$, we are done. If $\dim(V)=4$, by the same argument the totally geodesic $\mathsf{S}^{4}$ with $T_{o}\mathsf{S}^{4}=V$ is extrinsically homogeneous, contradicting the discussion above. Assume that $\dim(V)\geq 5$. Thus $\mathsf{K}^{\pi}$ acts trivially on $T_{o}\mathsf{S}^{7}\ominus V$, since $\mathsf{K}^{\pi}$ is simple and $\dim(T_{o}\mathsf{S}^{7}\ominus V)\leq 2$. Then $\mathsf{K}^{\pi}$ acts trivially on $T_{o}\mathsf{S}^{7}$, but this yields a contradiction that proves that $V=\pi$, and then we are done.

Case (ii): $\mathsf{K}^{\pi}$ acts non-trivially on $\pi$. Then, since $\mathsf{K}^{\pi}$ is simple, $\mathsf{K}^{\mathsf{\pi}}$ must act almost effectively on $\pi$. Then $\{k_{|\pi}:k\in\mathsf{K}^{\pi}\}=\mathsf{SO}(\pi)\cong\mathsf{SO}_{3}$. Assume that $\mathsf{S}^{3}$ is not homogeneous. Then, $\{k_{|T_{p}\mathsf{S}^{3}}:k\in\mathsf{K}^{T_{p}\mathsf{S}^{3}}\}=\mathsf{SO}(T_{p}\mathsf{S}^{3})$ for every, $p\in\mathsf{S}^{3}$. Otherwise, $\mathsf{K}^{T_{p}\mathsf{S}^{3}}$ would act trivially for some $p\in\mathsf{S}^{3}$, yielding a contradiction by the discussion in Case (i). This implies that $\mathsf{S}^{3}$ is extrinsically homogeneous. ∎