A Bonnet-Myers type theorem for quaternionic contact structures

TL;DR

This paper establishes a Bonnet-Myers type theorem for quaternionic contact manifolds of dimension greater than 7, linking Ricci-type bounds to compactness and diameter bounds in the sub-Riemannian setting.

Contribution

It extends classical Bonnet-Myers results to higher-dimensional quaternionic contact manifolds with new curvature bounds involving third-order derivatives.

Findings

Manifolds are compact under specified conditions.

Provides a sharp bound on sub-Riemannian diameter.

Establishes criteria for compactness in quaternionic contact geometry.

Abstract

We prove a Bonnet-Myers type theorem for quaternionic contact manifolds of dimension bigger than 7. If the manifold is complete with respect to the natural sub-Riemannian distance and satisfies a natural Ricci-type bound expressed in terms of derivatives up to the third order of the fundamental tensors, then the manifold is compact and we give a sharp bound on its sub-Riemannian diameter.

Equations

$$\mathrm{span}\{[X_{j_1},[X_{j_2},[\dots ,[X_{j_{m-1}},X_{j_m}]]]](x)\mid m\ge 1\}=T_{x}M,\forall x\in M,$$

$$d_{SR}(x,y)=\inf \{\ell (\gamma )\mid \gamma (0)=x,\gamma (T)=y,\gamma \text{ horizontal}\}.$$

$$\ell (\gamma )={\int }_0^T\sqrt{g(\dot{\gamma }(t),\dot{\gamma }(t))}dt.$$

$$I_{\alpha }{I}_{\beta }=-I_{\beta }{I}_{\alpha }=I_{\tau },I_{\alpha }^2=I_{\beta }^2=I_{\tau }^2=I_{\alpha }{I}_{\beta }{I}_{\tau }=-{\mathrm{id}}_{{|}_{\mathcal{D}}}.$$

$$g(I_{\alpha }X,I_{\alpha }Y)=g(X,Y),2g(I_{\alpha }X,Y)=d{\eta }_{\alpha }(X,Y).$$

$${\mathbb{S}}^3\hookrightarrow {\mathbb{S}}^{4n+3}\stackrel{\pi }{\to }{\mathbb{HP}}^n,n\ge 1.$$

$${\mathbb{R}}^{4n+3}={\mathbb{H}}^n\oplus \mathrm{Im}(\mathbb{H})$$

$$(z,w)\cdot (z^{\prime },w^{\prime })=\left(z+z^{\prime },w+w^{\prime }+\frac{1}{2}\mathrm{Im}(z{\bar{z}}^{\prime })\right).$$

$${\eta }_{\alpha }({\xi }_{\beta })={\delta }_{\alpha \beta },({\xi }_{\alpha }┘d{\eta }_{\alpha }{)}_{{|}_{\mathcal{D}}}=0,({\xi }_{\alpha }┘d{\eta }_{\beta }{)}_{{|}_{\mathcal{D}}}=-({\xi }_{\beta }┘d{\eta }_{\alpha }{)}_{{|}_{\mathcal{D}}},$$

$$2{\omega }_{\alpha |\mathcal{D}}=d{\eta }_{\alpha |\mathcal{D}},\xi ┘{\omega }_{\alpha }=0,\xi \in V.$$

$$T(X,Y)=-[X,Y{]}_{|V}=2\sum _{\alpha =1}^3{\omega }_{\alpha }(X,Y){\xi }_{\alpha }.$$

$$\mathrm{Ric}(X,X)-\sum _{\alpha =1}^{3}R(X,I_{\alpha }X,I_{\alpha }X,X)\ge 4(n-1)\kappa ,\forall X\in \mathcal{D}.$$

$$2n{T}^0(X,X)+(4n-8)U(X,X)+2(n-1)S\ge 4(n-1)\kappa ,\forall X\in \mathcal{D}.$$

$$T^0(\xi ,X,Y)=\frac{1}{2}{\mathcal{L}}_{\xi }g(X,Y),T^0(X,Y)=\frac{1}{2}\sum _{\alpha =1}^3({\mathcal{L}}_{{\xi }_{\alpha }}g)(I_{\alpha }X,Y).$$

$$U(X,Y)=$$

$$+\frac{1}{2n}\sum _{i=1}^{4n}g(({\mathcal{L}}_{{\xi }_{\beta }}{I}_{\alpha })I_{\tau }{X}_i,X_i)g(X,Y),$$

$$S=d{\eta }_{\alpha }({\xi }_{\beta },{\xi }_{\tau })-d{\eta }_{\tau }({\xi }_{\alpha },{\xi }_{\beta })-d{\eta }_{\beta }({\xi }_{\tau },{\xi }_{\alpha })-\frac{1}{2n}\sum _{i=1}^{4n}g(({\mathcal{L}}_{{\xi }_{\beta }}{I}_{\alpha })I_{\tau }{X}_i,X_i).$$

$$T^0=U=0,S=2.$$

$$\sum _{\alpha =1}^{3}R(X,I_{\alpha }X,I_{\alpha }X,X)=\sum _{\alpha =1}^3{R}^g(X,I_{\alpha }X,I_{\alpha }X,X)+9=12,$$

$${\mathcal{L}}_X:\mathcal{D}\to TM/\mathcal{D},{\mathcal{L}}_X(Y):=[X,Y]\mathrm{mod}\mathcal{D}$$

$$g(I_{\tau }X,X)=d\eta (X,X)=0.$$

$$g(I_{\alpha }X,I_{\beta }X)=-g(X,I_{\alpha }{I}_{\beta }X)=-g(X,I_{\tau }X)=0.$$

$${\Omega }_{\alpha \beta }={\eta }_{\alpha }({\mathcal{L}}_X(I_{\beta }X))=-d{\eta }_{\alpha }(X,I_{\beta }X)=-2g(I_{\alpha }X,I_{\beta }X)=-2{\delta }_{\alpha \beta }g(X,X).\text{hfill}\text{qed}$$

$$H(\lambda ):=\frac{1}{2}\sum _{i=1}^k⟨\lambda ,X_i{⟩}^2,\lambda \in T^{\ast }M,$$

$$\dot{\lambda }(t)=\vec{H}(\lambda (t)).$$

$$\dot{V}(t):={\frac{d}{d\epsilon }|}_{\epsilon =0}{e}_{\ast }^{-\epsilon \vec{H}}V(t+\epsilon ).$$

$$\dot{\mathcal{J}}=0.$$

$${\mathcal{V}}_{\lambda }:=\ker {\pi }_{\ast }{|}_{\lambda }=T_{\lambda }(T_{\pi (\lambda )}^{\ast }M)\subset T_{\lambda }(T^{\ast }M).$$

$$\mathcal{J}(t)=\sum _{i=1}^d{p}_i(t)E_i(t)+x_i(t)F_i(t).$$

$$\frac{d}{dt}\left(\begin{array}{c}E\\ F\end{array}\right)=\left(\begin{array}{cc}\mathcal{A}(t)& -\mathcal{B}(t)\\ \mathcal{R}(t)& -\mathcal{A}(t{)}^{\ast }\end{array}\right)\left(\begin{array}{c}E\\ F\end{array}\right),$$

$$\sigma (E_i,E_j)=\sigma (F_i,F_j)=\sigma (E_i,F_j)-{\delta }_{ij}=0,i,j=1,\dots ,d.$$

Full text

1. 1 Introduction and main results
2. 2 Curvature of sub-Riemannian qc structures
3. 3 Structural equations for the coordinate frame
4. 4 Symplectic products and canonical frame
5. 5 Proof of Theorems 1 and 2
6. A Some technical facts and useful identities

1. Introduction and main results

Bonnet-Myers theorem is classical among comparison theorems in Riemannian geometry [37]. It states that, if the Ricci curvature of a complete $d$-dimensional Riemannian manifold $(M,g)$ is bounded below by $(d-1)\kappa>0$, then the manifold $M$ is compact and its diameter is at most $\pi/\sqrt{\kappa}$.

Several generalizations of this theorem, in variuos smooth settings (and even in the non-smooth one of metric measure spaces, see for instance [38]) have been recently investigated, introducing suitable notion of curvature or Ricci bound. Among these, different versions of Bonnet-Myers theorem have been obtained in the setting of sub-Riemannian geometry (cf. discussion in Section 1.4).

Recall that a sub-Riemannian structure $(\mathcal{D},g)$ on a smooth, connected manifold $M$ of dimension $d\geq 3$ is defined by a vector distribution $\mathcal{D}$ of constant rank $k\leq d$ and a smooth metric $g$ assigned on $\mathcal{D}$. The distribution is required to satisfy the Hörmander condition, or to be bracket-generating, that means

[TABLE]

for some (and then any) set $X_{1},\ldots,X_{k}\in\Gamma(\mathcal{D})$ of local generators for $\mathcal{D}$.

Given a sub-Riemannian structure on $M$, the sub-Riemannian distance is defined by:

[TABLE]

where a Lipschitz continuous path $\gamma:[0,T]\to\mathbb{R}$ is horizontal if it satisfies $\dot{\gamma}(t)\in\mathcal{D}_{\gamma(t)}$ for almost every $t$, and in this case we set

[TABLE]

By the classical Chow-Rashevskii theorem (see for instance [6, Chapter 3]), the condition (1) implies that $d_{SR}$ is finite and continuous on $M\times M$. We say that the sub-Riemannian manifold is complete if $(M,d_{SR})$ is complete as a metric space.

A sub-Riemannian Bonnet-Myers theorem states, under suitable curvature conditions, that the manifold $M$ is compact and gives a bound on its sub-Riemannian diameter. For more details on sub-Riemannian geometry we refer to classical references such as [16, 36] and the more recent ones [6, 35, 39].

Remark 1*.*

Notice that if the sub-Riemannian structure is defined as the restriction of a Riemannian metric $g$ on $M$ to a distribution $\mathcal{D}$, in general the sub-Riemannian diameter is bigger than the Riemannian one. Thus, even if one is able to control the Riemannian curvature of $(M,g)$ and apply a classical Bonnet-Myers theorem, one can prove compactness of $M$, but has no a priori estimate on the sub-Riemannian diameter.

In this paper we focus on quaternionic contact structure. A quaternionic contact (qc) structure, introduced in [17], appears naturally as the conformal boundary at infinity of the quaternionic hyperbolic space. The qc structure gives a natural geometric setting for the quaternionic contact Yamabe problem, [23, 43, 31, 29]. A particular case of this problem amounts to find the extremals and the best constant in the $L^{2}$ Folland-Stein Sobolev-type embedding, [21] and [22], with a complete description of the extremals and the best constant on the quaternionic Heisenberg groups [29, 30, 27].

A quaternionic contact structure carry a natural sub-Riemannian structure with a codimension three distribution. Curvature conditions are expressed in terms on bounds on standard curvature tensors of quaternionic contact geometry. These conditions can be expressed only in terms of sub-Riemannian quantities (cf. Theorems 1 and 2) and are obtained through the computation of the sub-Riemannian coefficients of the generalized Jacobi equation, first introduced in [9, 44] and subsequently developed in [10, 11, 2].

1.1. Quaternionic contact structure

A quaternionic contact manifold $(M,\mathbb{Q},g)$ is a $(4n+3)$-dimensional manifold $M$ with a codimension-three distribution $\mathcal{D}$ equiped with $\mathrm{Sp}(n)\mathrm{Sp}(1)$ structure. Explicitly, the distribution $\mathcal{D}$ is locally described as the kernel of a 1-form $\eta=(\eta_{1},\eta_{2},\eta_{3})$ with values in $\mathbb{R}^{3}$ together with a compatible Riemannian metric $g$ and a rank-three bundle $\mathbb{Q}$ consisting of endomorphisms of $\mathcal{D}$ locally generated by three almost complex structures $I_{1},I_{2},I_{3}:\mathcal{D}\to\mathcal{D}$ satisfying the identities of the imaginary unit quaternions. Namely, if $\{\alpha,\beta,\tau\}$ is any cyclic permutation of $\{1,2,3\}$ we have

[TABLE]

Moreover $I_{1},I_{2},I_{3}$ are compatible with the metric $g$, in the following sense: for every $\alpha=1,2,3$ and $X,Y\in\mathcal{D}$ we have

[TABLE]

From the sub-Riemannian view-point, these structures are fat, i.e. for any non zero section $X$ of $\mathcal{D}$, $TM$ is (locally) generated by $\mathcal{D}$ and $[X,\mathcal{D}]$. This is a direct consequence of the quaternionic relations of the almost complex structures. For completeness a proof is given in Section 2. The fat condition is open in the $C^{1}$ topology, however it gives some restriction on the rank $k$ of the distribution (for example $\dim M\leq 2k-1$, [36, Prop. 5.6.3]).

Example 1* (Quaternionic Hopf fibration).*

A classical example of quaternionic contact structure is the quaternionic Hopf fibration

[TABLE]

Here $\mathcal{D}=(\ker\pi_{*})^{\perp}$ is the orthogonal complement of the kernel of the differential of the Hopf map $\pi$, and the sub-Riemannian metric is the restriction to $\mathcal{D}$ of the standard round metric on $\mathbb{S}^{4n+3}$. The sub-Riemannian distance on the quaternionic Hopf fibration can be computed explicitly and its diameter is $\pi$, as it is proved in [15]. This example is one of the simplest (non-Carnot) sub-Riemannian structures of corank greater than $1$, and is included in the sub-class of $3$-Sasakian structures.

Example 2* (Quaternionic Heisenberg group).*

An example of quaternionic contact structure that is not 3-sasakian is the quaternionic Heisenberg group. It is defined as

[TABLE]

endowed with the group law

[TABLE]

If we take $\mathcal{D}=\mathbb{H}^{n}$ (which has dimension $4n$) with the standard Euclidean metric, it easy to see that it is bracket generating and defines a quaternionic contact structure.

1.2. Biquard connection, torsion and curvature

On a qc manifold of dimension $4n+3$ with $n\geq 2$ with a fixed metric $g$ on the horizontal distribution $\mathcal{D}$ there exists a canonical connection, called Biquard connection, defined in [17]. Biquard shows that there exists a unique supplementary subspace $V$ to $\mathcal{D}$ in $TM$ and a unique connection $\nabla$ with torsion $T$, such that:

• (i)

$\nabla$ preserves the decomposition $H\oplus V$ and the $\mathrm{Sp}(n)\mathrm{Sp}(1)$ structure on $\mathcal{D}$, $\nabla g=0$, $\nabla\sigma\in\Gamma(\mathbb{Q})$ for $\sigma\in\Gamma(\mathbb{Q})$, and its torsion on $\mathcal{D}$ is given by $T(X,Y)=-[X,Y]_{|V}$;

• (ii)

for $\xi\in V$, the endomorphism $T(\xi,.)_{|\mathcal{D}}$ of $\mathcal{D}$ lies in111the perpendicular is computed with respect to the inner product $\langle A\,|\,B\rangle={\ \sum_{i=1}^{4n}g(A(e_{i}),B(e_{i})),}$ for $A,B\in\text{End}(H)$. $(\mathrm{sp}(n)\oplus\mathrm{sp}(1))^{\bot}\subset\mathrm{gl}(4n)$;

• (iii)

the connection on $V$ is induced by the natural identification $\varphi$ of $V$ with the subspace $\mathrm{sp}(1)$ of the endomorphisms of $\mathcal{D}$, i.e., $\nabla\varphi=0$.

When the dimension of $M$ is at least eleven, [17] shows that the supplementary vertical distribution $V$ is (locally) generated by three Reeb vector fields $\xi_{1},\xi_{2},\xi_{3}$ determined by the conditions

[TABLE]

where $\lrcorner$ denotes the interior multiplication: more explicitly $X\lrcorner\Phi=\Phi(X,\cdot)$ where $X$ is a vector field and $\Phi$ is a differential 2-form.

Remark 2*.*

In this paper we restrict our attention to quaternionic contact structure of dimension strictly bigger that seven. If the dimension of $M$ is seven Duchemin shows in [20] that if we assume, in addition, the existence of Reeb vector fields as in (4), then the Biquard result holds. Henceforth, by a qc structure in dimension $7$ we shall mean a qc structure satisfying (4). This implies the existence of the connection with properties (i), (ii) and (iii) above.

The fundamental 2-forms $\omega_{\alpha}$ of the qc structure are defined by

[TABLE]

The torsion restricted to $\mathcal{D}$ has the form

[TABLE]

The properties of the Biquard connection are encoded in the torsion endomorphism $T(\xi,.)_{|\mathcal{D}}$. It is completely trace-free, $\mathrm{tr}(T(\xi,.)_{|\mathcal{D}})=\mathrm{tr}T(\xi,.)_{|\mathcal{D}}\circ I_{\alpha}=0$ and can be decomposed into symmetric and skew-symmetric parts, $T(\xi_{\alpha},.)_{|\mathcal{D}}=T^{0}(\xi_{\alpha},.)_{|\mathcal{D}}+I_{\alpha}u$, respectively where $u$ is a traceless symmetric (1,1)-tensor on $\mathcal{D}$ which commutes with $I_{1},I_{2},I_{3}$, see [17]. When $n=1$ the tensor $u$ vanishes identically and the torsion is a symmetric tensor, $T_{\xi}=T_{\xi}^{0}$.

The two $\mathrm{Sp}(n)\mathrm{Sp}(1)$-invariant trace-free symmetric 2-tensors $T^{0},U$ on $\mathcal{D}$ defined in (33) having the properties (34) determine completely the symmetric and the skew-symmetric parts of torsion endomorphism, respectively [31] (cf. (35) and (36) in the Appendix.)

The qc-Ricci tensor $\mathrm{Ric}$ and the normalized qc-scalar curvature $S$ of the Biquard connection are defined with the usual horizontal traces of the curvature of the Biquard connection (cf.  (37) in the Appendix.)

A qc structure is said to be qc-Einstein if the horizontal qc-Ricci tensor is a scalar multiple of the metric. As shown in [31, 28] the qc-Einstein condition is equivalent to the vanishing of the torsion endomorphism of the Biquard connection. In this case $S$ is constant and the vertical distribution is integrable. It is also worth recalling that the horizontal qc-Ricci tensors and the integrability of the vertical distribution can be expressed in terms of the torsion of the Biquard connection according to (38) in the Appendix (see [31], cf. also [29, 33, 32]) .

Any 3-Sasakian manifold has zero torsion endomorphism, and the converse is locally true if in addition the qc scalar curvature is a positive constant [31].

1.3. Main results

Our main result reads as follows, in terms of the qc Ricci tensor and the curvature tensor associated with the Biquard connection.

Theorem 1**.**

Let $(M,g,\mathbb{Q})$ be a $4n+3$-dimensional complete qc manifold with $n>1$. Assume that there exists a constant $\kappa>0$ such that

[TABLE]

Then $(M,g,\mathbb{Q})$ is compact manifold with finite fundamental group, and its sub-Riemannian diameter is not greater than $\pi/\sqrt{\kappa}$.

Remark 3*.*

The bound on the sub-Riemannian diameter given in Theorem 1 is sharp since the equality is attained for the quaternionic Hopf fibration, where $\kappa$ can be chosen equal to 1 and the sub-Riemannian diameter is $\pi$ (cf. Example 1 and Remark 5). Moreover, since $R(X,X,X,X)=0$, the left hand side in (5) is indeed a trace on a $4(n-1)$-dimensional subspace of $\mathcal{D}$.

Theorem 1 can be also restated as follows, in terms of the horizontal part of torsion tensors and scalar curvature.

Theorem 2**.**

Let $(M,g,\mathbb{Q})$ be a $4n+3$-dimensional complete qc manifold with $n>1$. Assume that there exists a constant $\kappa>0$ such that

[TABLE]

Then $(M,g,\mathbb{Q})$ is compact manifold with finite fundamental group, and its sub-Riemannian diameter not greater than $\pi/\sqrt{\kappa}$.

The proofs of Theorems 1 and 2 are given in Section 5. We stress that, thanks to the results in [31] and the proof of [34, Theorem 4.2.5], the condition (6) can be rewritten only in terms of the qc structure and its Lie derivatives.

Proposition 3**.**

Let $\{X_{1},\ldots,X_{4n}\}$ be a local orthonormal basis for $\mathcal{D}$ and $\{\alpha,\beta,\tau\}$ be any cyclic permutation of $\{1,2,3\}$. We have the following relations:

• (i)

The symmetric part of the torsion endomorphism is determined entirely by the Lie derivative of the metric

[TABLE]

• (ii)

The skew-symmetric part of the torsion described by $U$ satisfies

[TABLE]

• (iii)

The normalized qc scalar curvature is written as

[TABLE]

Remark 4*.*

We note that Theorems 1 and 2 generalize Bonnet-Myers results for the sub-class of qc manifold with integrable vertical space obtained in [24] simplifying considerably the Bonnet-Myers positivity condition and giving moreover explicit diameter bounds.

Remark 5* (3-Sasakian case).*

Assume that the qc manifold is 3-Sasakian. In this case, we have from [31, Corollary 4.13] and [31, Theorem 3.12] that

[TABLE]

Therefore, (6) is satisfied with $\kappa=1$ and we recover the universal diameter bound for 3-Sasakian manifold, established in [40]. The diameter bound is attained for the quaternionic Hopf fibration (cf. Example 1).

Notice that in [40] the authors use curvature tensors $R^{g}$ associated with the Levi-Civita connection. Using the relation between $R^{g}$ and the curvature tensor $R$ associated with the Biquard connection (see [31, Corollary 4.13]) we have

[TABLE]

where we apply the identity $\sum_{\alpha=1}^{3}R^{g}(X,I_{\alpha}X,I_{\alpha}X,X)=3$, valid for 3-Sasakian manifolds, and proved in [42, Prop. 3.2].

We also state the following interesting corollary of Theorem 2, when $n=2$.

Corollary 4**.**

Let $(M,g,\mathbb{Q})$ be a $11$-dimensional complete qc manifold. Assume $T^{0}=0$ and $S\geq 2\kappa>0$, then $M$ is compact with sub-Riemannian diameter not greater than $\pi/\sqrt{\kappa}$.

We note that qc manifolds with $T^{0}=0$ are characterized with the condition that the almost contact structure on the corresponding twistor space is normal, see [19].

1.4. Relation with previous literature

Other sub-Riemannian Bonnet-Myers type results are found in the literature, proved with different techniques and for different sub-Riemannian structures. The three dimensional contact case has been considered using second variation like formulas in [41] (for CR structures) and in [25]. In [12] and [13], a version of Bonnet-Myers has been proved using heat semigroup approach for Yang-Mills type structures with transverse symmetries and Riemannian foliations with totally geodesic leaves, respectively. Using Riccati comparison techniques for 3D contact [7], 3-Sasakian [40] and general contact sub-Riemannian structure [1]. See also [10] for a general approach to sub-Riemannian Bonnet-Myers theorem through curvature invariants.

A compactness result, obtained by applying Riemannian classical Bonnet-Myers theorem to a suitable Riemannian extension of the metric (cf. Remark 1) is obtained in [14] for contact manifolds and in [24] for quaternionic contact manifolds with integrable vertical space.

1.5. Structure of the paper

In Section 2 we recall some results about the sub-Riemannian Jacobi equation and the curvature invariants. In Sections 3 and 4 we carefully compute these invariants for quaternionic contact structures and express them with respect to standard tensors of quaternionic contact geometry. To perform these computations, we introduce a generalized Fermi frame along the geodesic. In Section 5 we use these computations to prove the Bonnet-Myers theorem. Appendix A resumes geometric properties of quaternionic contact structures.

2. Curvature of sub-Riemannian qc structures

In this section we resume the basic facts on sub-Riemannian geodesic flows and curvature needed to prove our results. For a more comprehensive presentation we refer the reader to [2, 10, 11].

2.1. Quaternionic contact sub-Riemannian structures are fat

A sub-Riemannian structure is said to be fat if for any non zero section $X$ of $\mathcal{D}$, $TM$ is (locally) generated by $\mathcal{D}$ and $[X,\mathcal{D}]$. This is equivalent to show that for every non zero horizontal vector $X\in\mathcal{D}$ the following map is surjective

[TABLE]

Notice that the map $\mathcal{L}_{X}$ is tensorial, in the sense that for each $x\in M$ the value of $[X,Y](x)\mod\mathcal{D}_{x}$ depends only on $X(x)$ and $Y(x)$. Moreover $\dim T_{x}M/\mathcal{D}_{x}=3$. The fat property follows from the following linear algebra observation.

Lemma 5**.**

The vectors $\{\mathcal{L}_{X}(I_{\alpha}X)\}_{\alpha=1,2,3}$ are linearly independent in $T_{x}M/\mathcal{D}_{x}$.

Proof.

Let us start by showing that, for every non zero horizontal vector $X\in\mathcal{D}$ the four vectors $X,I_{1}X,I_{2}X,I_{3}X$ are mutually orthogonal. First notice that for every $\tau=1,2,3$ and every horizontal vector $X\in\mathcal{D}$ one has

[TABLE]

Moreover, if $\{\alpha\beta\tau\}$ is a cyclic permutation of $\{123\}$, thanks to (2) and (7), one has for $\alpha\neq\beta$

[TABLE]

To prove that the sub-Riemannian structure is fat it is sufficient to show that the image through $L_{X}$ of the vectors $I_{\beta}X$ for $\beta=1,2,3$ is a basis of $T_{x}M/\mathcal{D}_{x}$, for every $x\in M$. This is equivalent to say that for every $X$ the matrix $\Omega_{\alpha\beta}=\eta_{\alpha}(L_{X}(I_{\beta}X))$ is invertible, which follows from

[TABLE]

2.2. Sub-Riemannian geodesic flow

Sub-Riemannian geodesics are horizontal curves that are locally minimizers for the length (between curve with same endpoints). The sub-Riemannian Hamiltonian $H:T^{*}M\to\mathbb{R}$ is defined as

[TABLE]

where $X_{1},\ldots,X_{k}$ is any local orthonormal frame for $\mathcal{D}$ and $\langle\lambda,v\rangle$ denotes the action of a covector $\lambda\in T_{x}^{*}M$ on a vector $v\in T_{x}M$, based at $x\in M$. Let $\sigma$ be the canonical symplectic form on $T^{*}M$. The Hamiltonian vector field $\vec{H}$ is defined by the identity $\sigma(\cdot,\vec{H})=dH$. Then the Hamilton equations are

[TABLE]

Solutions of (8) are called extremals, and one can prove that their projections $\gamma(t):=\pi(\lambda(t))$ on $M$ are geodesics [6, Chapter 4]. The Hamiltonian $H$ is constant along an extremal $\lambda(t)$ and we say that the extremal is length-parametrized if $H(\lambda(t))=1/2$.

Since the sub-Riemannian structure defined on the qc manifold is fat, any minimizer can be recovered uniquely in this way. This statement is not true in full generality, since there can exist minimizing trajectories might not satisfy the Hamiltonian equation  (8). These trajectories are called abnormal minimizers and are related to the main open problems in sub-Riemannian geometry (see for instance [5] for a discussion).

2.3. Jacobi equation revisited

Given an extremal $\lambda(t)$ of the sub-Riemannian Hamiltonian flow and a vector field $V(t)$ along $\lambda(t)$ we define

[TABLE]

the Lie derivative of $V$ in the direction of $\vec{H}$. A vector field $\mathcal{J}(t)$ along $\lambda(t)$ is a sub-Riemannian Jacobi field if

[TABLE]

If $M$ has dimension $d$, the set of solutions of (9) is a $2d$-dimensional vector space. The projections $\pi_{*}\mathcal{J}(t)$ are vector fields on the manifold $M$ corresponding to one-parameter variations of $\gamma(t)=\pi(\lambda(t))$ through geodesics. In the Riemannian case, this coincides with the classical construction of Jacobi fields.

Next, let us write (9) using the symplectic structure $\sigma$ of $T^{*}M$. Observe that on $T^{*}M$ there is a natural notion of vertical subspace at $\lambda\in T^{*}M$, namely

[TABLE]

Then $\mathcal{V}$ is a smooth (Lagrangian) sub-bundle of $T(T^{*}M)$. If one considers the frame $E_{i}=\partial_{p_{i}}|_{\lambda(t)}$, and $F_{j}=\partial_{x_{j}}|_{\lambda(t)}$ induced by coordinates $(x_{1},\ldots,x_{d})$ on $M$, then the vector field $\mathcal{J}(t)$ has components $(p(t),x(t))\in\mathbb{R}^{2d}$, that means

[TABLE]

and the elements of the frame satisfy the equation

[TABLE]

for some smooth families of $d\times d$ matrices $\mathcal{A}(t),\mathcal{B}(t),\mathcal{R}(t)$, where $\mathcal{B}(t)=\mathcal{B}(t)^{*}$ and $\mathcal{R}(t)=\mathcal{R}(t)^{*}$. The structure of (10) follows from the fact that the frame is Darboux, namely

[TABLE]

The idea is then to look for a suitable Darboux frame $\{E_{i}(t),F_{i}(t)\}_{i=1}^{d}$ along $\lambda(t)$ such that the equations above are in normal form.

2.4. Curvature coefficients in quaternionic contact

The normal form of the sub-Riemannian Jacobi equation (10) has been first studied by Agrachev-Zelenko in [3, 4] and subsequently completed by Zelenko-Li in [44]. In particular, there exist a normal form of (10) where the matrices $\mathcal{A}(t)$ and $\mathcal{B}(t)$ are constant. Here we give an ad-hoc statement for quaternionic contact sub-Riemannian structures, following the notation and the presentation of [11].

Remark 6*.*

It is convenient to split the set of indices $1,\ldots,4n+3$ into three subsets $a,b,c$ with cardinality $|a|=|b|=3$ and $|c|=4n-3$. The index $a$ parametrizes the three-dimensional complement to the distribution, while $b$ and $c$ together parametrize the set of indices on the distribution.

This splitting is related to the fact that the Lie derivative $\mathcal{L}_{X}:\mathcal{D}_{x}\to T_{x}M/\mathcal{D}_{x}$ in the direction of a nontrivial horizontal vector $X\in\mathcal{D}_{q}$ induces a well defined, surjective linear map with $3$-dimensional image (the “$a$” space) and a $4n-3$-dimensional kernel (the “$c$” space). The orthogonal complement of the kernel within $\mathcal{D}_{x}$ is a $3$-dimensional space (the “$b$” space).

Accordingly to this decomposition, any $(4n+3)\times(4n+3)$ matrix $L$ is written in the block form

[TABLE]

with similar notation for row or column vectors.

Theorem 6**.**

Let $\lambda(t)$ be a sub-Riemannian extremal of a qc sub-Riemannian structure. There exists a smooth moving frame along $\lambda(t)$

[TABLE]

such that the following holds true for any $t$:

• (i)

$\operatorname{\mathrm{span}}\{E_{a}(t),E_{b}(t),E_{c}(t)\}=\mathcal{V}_{\lambda(t)}$.

• (ii)

It is a Darboux basis, namely

[TABLE]

• (iii)

The frame satisfies the structural equations

[TABLE]

where the curvature matrix $\mathcal{R}(t)=\mathcal{R}(t)^{*}$ is

[TABLE]

and satisfies the additional condition $\mathcal{R}_{ab}(t)=-\mathcal{R}_{ab}(t)^{*}$.

Remark 7*.*

If we fix another frame $\{\widetilde{E}(t),\widetilde{F}(t)\}$ satisfying (i)-(iii) for some matrix $\widetilde{\mathcal{R}}(t)$, then there exists a constant $n\times n$ orthogonal matrix $O$ that preserves the structural equations and such that

[TABLE]

For more details about the uniqueness of this frame we refer the reader to [11].

2.5. Ricci curvature and Bonnet-Myers theorem

We can state now a corollary of the general results obtained in [10] (an analogue statement of the one mentioned here is [40, Thm. 5]) that gives a Bonnet-Myers type theorem that we will use to prove our results.

Theorem 7**.**

Let $(M,\mathcal{D},g)$ be a complete qc sub-Riemannian manifold. Assume that there exists $\kappa>0$ such that for any length-parametrized extremal $\lambda(t)$ one has

[TABLE]

Then $M$ is compact and its sub-Riemannian diameter is bounded by $\pi/\sqrt{\kappa}$. Moreover $M$ has finite fundamental group.

In the following sections we will compute the quantity $\operatorname{\mathrm{tr}}(\mathcal{R}_{cc}(t))$ for every sub-Riemannian extremal on a qc manifold and deduce the main theorems stated in the Introduction.

3. Structural equations for the coordinate frame

In what follows latin indices $i,j,k,\dots$ belong to $\{1,\dots,4n\}$ and Greek ones $\alpha,\beta,\tau,\ldots$ belong to $\{1,2,3\}$, corresponding to quaternions (following the same quaternionic indices notation of Appendix A).

We start by choosing a convenient local frame on $M$, associated with a given trajectory. Here $\{X_{1},\ldots,X_{4n}\}$ will denote a local orthonormal frame for the metric $g$ on $\mathcal{D}$.

3.1. Fermi frame

Given a geodesic $\gamma(t)=\pi(\lambda(t))$, we define a convenient local frame on $M$ which is an application of a standard result in differential geometry, called Fermi normal frame along a smooth curve.

Lemma 8**.**

Given a geodesic $\gamma(t)$, there exists a $\mathbb{Q}$-orthonormal frame, i.e., a horizontal frame $X_{i}$, $i\in\{1,\dots,4n\}$, and vertical frame $\xi_{\alpha}$, $\alpha=1,2,3$ in a neighborhood of $\gamma(0)$, such that for all $\alpha,\beta\in\{1,2,3\}$ and $i,j\in\{1,\dots,4n\}$,

• (i)

the frame is orthonormal for the Riemannian metric $g+\sum_{\beta}\eta_{\beta}^{2}$,

• (ii)

$\nabla_{X_{i}}X_{j}|_{\gamma(t)}=\nabla_{\xi_{\alpha}}X_{j}|_{\gamma(t)}=\nabla_{X_{i}}\xi_{\beta}|_{\gamma(t)}=\nabla_{\xi_{\alpha}}\xi_{\beta}|_{\gamma(t)}=0$.

In particular, for all $\alpha,\beta,\tau\in\{1,2,3\}$ and $i,j\in\{1,\dots,4n\}$

[TABLE]

The proof of this Lemma is postponed to Appendix A.5.

3.2. Commutator relations and Poisson brackets

Fix $\{X_{1},X_{2},\dots$ $,X_{4n}\}$ a horizontal frame and $\xi_{\alpha}$ for $\alpha=1,2,3$ vertical frame and introduce the momentum functions $u_{i},v_{\alpha}:T^{*}M\to\mathbb{R}$ defined by

[TABLE]

The momentum functions define coordinates $(u,v)$ on each fiber of $T^{*}M$. In turn, they define local vector fields $\partial_{v_{\alpha}}$ and $\partial_{u_{i}}$ on $T^{*}M$ (satisfying $\pi_{*}\partial_{v_{\alpha}}=\pi_{*}\partial_{u_{i}}=0$). Moreover, they define also the Hamiltonian vector fields $\vec{u}_{i}$ and $\vec{v}_{\alpha}$. The Hamiltonian frame associated with $\{\xi_{\alpha},X_{i}\}$ is the local frame on $T^{*}M$ around $\lambda(0)$ given by $\{\partial_{u_{i}},\partial_{v_{\alpha}},\vec{u}_{i},\vec{v}_{\alpha}\}$.

The sub-Riemannian Hamiltonian and the corresponding Hamiltonian vector field are

[TABLE]

We will use the short notation $\alpha\beta=\tau$, where $\alpha,\beta=1,2,3$, for the quaternionic multiplication. The following $3\times 3$ skew-symmetric matrix contains the vertical part of the covector:

[TABLE]

with the convention $v_{\alpha^{2}}=-v_{1}=0$ which is the standard identification $\mathbb{R}^{3}\backsimeq\mathfrak{so}(3)$.

Remark 8*.*

For functions $f,g\in C^{\infty}(T^{*}M)$, the symbol $\{f,g\}$ denotes their Poisson bracket. The symbol $\dot{f}$ always denotes the Lie derivative in the direction of $\vec{H}$. We make systematic use of symplectic calculus (see for instance [8] for reference).

In what follows, we fix a geodesic $\gamma(t)$ with corresponding lift $\lambda(t)$ and a Fermi frame associated with it and given by Lemma 8. Repeated indices are implicitly summed over.

Lemma 9** (Commutators).**

We have the following identities

• (a’)

$g([X,Y],\xi_{\alpha})=-d\eta_{\alpha}(X,Y)=-2g(I_{\alpha}X,Y)$**

• (b’)

$g([X,Y],Z)=g(\nabla_{X}Y,Z)-g(\nabla_{Y}X,Z)$**

• (c’)

$g([\xi_{\alpha},X],\xi_{\beta})=d\eta_{\alpha}(\xi_{\beta},X)=-d\eta_{\beta}(\xi_{\alpha},X)=-g(\nabla_{X}\xi_{\alpha},\xi_{\beta}).$**

• (d’)

$g([\xi_{\alpha},X],Y)=-T(\xi_{\alpha},X,Y)+g(\nabla_{\xi_{\alpha}}X,Y)$**

• (e’)

$g([\xi_{\alpha},\xi_{\beta}],\xi_{\gamma})=-d\eta_{\gamma}(\xi_{\alpha},\xi_{\beta})=Sg(\xi_{\alpha\beta},\xi_{\gamma})+g(\nabla_{\xi_{\alpha}}\xi_{\beta},\xi_{\gamma})-g(\nabla_{\xi_{\beta}}\xi_{\alpha},\xi_{\gamma})$**

• (f’)

$g([\xi_{\alpha},\xi_{\beta}],X)=\rho_{\alpha\beta}(I_{\beta}X,\xi_{\beta})=\rho_{\alpha\beta}(I_{\alpha}X,\xi_{\alpha})$**

Along the curve we have the following simplifications

• (a)

$g([X,Y],\xi_{\alpha})=-d\eta_{\alpha}(X,Y)=-2g(I_{\alpha}X,Y)$**

• (b)

$g([X,Y],Z)=0$**

• (c)

$g([\xi_{\alpha},X],\xi_{\beta})=0$**

• (d)

$g([\xi_{\alpha},X],Y)=-T(\xi_{\alpha},X,Y)$**

• (e)

$g([\xi_{\alpha},\xi_{\beta}],\xi_{\gamma})=-d\eta_{\gamma}(\xi_{\alpha},\xi_{\beta})=Sg(\xi_{\alpha\beta},\xi_{\gamma})$**

• (f)

$g([\xi_{\alpha},\xi_{\beta}],X)=\rho_{\alpha\beta}(I_{\beta}X,\xi_{\beta})$**

Proof.

It is obtained by a direct computation by combining the definition and the properties of the torsion of the Biquard connection (38) and using the choice of Fermi frame. ∎

As a consequence of the previous identities we compute the following Poisson brackets of momentum functions.

Lemma 10** (Poisson brackets).**

The momentum functions $u_{i},v_{\alpha}$ have the following properties:

• (a’)

$\{v_{\alpha},u_{i}\}=d\eta_{\alpha}(\xi_{\tau},X_{i})v_{\tau}+\left(-T(\xi_{\alpha},X_{i},X_{k})+g(\nabla_{\xi_{\alpha}}X_{i},X_{k})\right)u_{k}$,

• (b’)

$\{v_{\alpha},v_{\beta}\}=-d\eta_{\tau}(\xi_{\alpha},\xi_{\beta})v_{\tau}+\rho_{\alpha\beta}(I_{\beta}X_{k},\xi_{\beta})u_{k}$,

• (c’)

$\{u_{i},u_{j}\}=-2g(I_{\tau}X_{i},X_{j})v_{\tau}+g(X_{k},[X_{i},X_{j}])u_{k}$.

Moreover, when evaluated along the extremal $\lambda(t)$, one has

• (a)

$\{v_{\alpha},u_{i}\}=-T(\xi_{\alpha},X_{i},\dot{\gamma})$,

• (b)

$\{v_{\alpha},v_{\beta}\}=-d\eta_{\tau}(\xi_{\alpha},\xi_{\beta})v_{\tau}+\rho_{\alpha\beta}(I_{\beta}\dot{\gamma},\xi_{\beta})=Sg(\xi_{\alpha\beta},\xi_{\tau})v_{\tau}+\rho_{\alpha\beta}(I_{\beta}\dot{\gamma},\xi_{\beta})$,

• (c)

$\{u_{i},u_{j}\}=-2g(I_{\tau}X_{i},X_{j})v_{\tau}$.

• (d)

$\partial_{u_{k}}\{u_{i},u_{j}\}=0$,

• (e)

$\partial_{v_{\alpha}}\{u_{i},u_{j}\}=-2g(I_{\alpha}X_{i},X_{j})$,

Proof.

These formulas comes as a direct consequence of Lemma 9, thanks to the following observation: let $X,Y$ be two smooth vector fields on $M$ and $h_{X}(\lambda)=\langle\lambda,X(x)\rangle$ and $h_{Y}(\lambda)=\langle\lambda,Y(x)\rangle$ the associated Hamiltonians that are linear on fibers (here $x=\pi(\lambda)$). Then we have the identity $\{h_{X},h_{Y}\}(\lambda)=\langle\lambda,[X,Y](x)\rangle=:h_{[X,Y]}$.

Let us now prove, as an example, formula (c’). We have, by definition

[TABLE]

Using then the above observation one has

[TABLE]

and using then Lemma 9, one gets

[TABLE]

which proves (c’). Observe that the last term vanishes when evaluated along the extremal, thanks to the properties of Fermi frame. This proves (c). All other formulas follow analogously. ∎

Lemma 11** (Some arrows).**

We have the following expressions along the extremal

• (a)

$\overrightarrow{\{u_{i},u_{j}\}}=-2g(I_{\alpha}X_{i},X_{j})\vec{v}_{\alpha}-u_{k}X_{\ell}g([X_{i},X_{j}],X_{k})\partial_{u_{\ell}}-u_{k}\xi_{\beta}g([X_{i},X_{j}],X_{k})\partial_{v_{\beta}}$,

• (b)

$\overrightarrow{\{v_{\alpha},u_{i}\}}=-T(\xi_{\alpha},X_{i},X_{k})\vec{u}_{k}-K_{\alpha i}^{\ell}\partial_{u_{\ell}}-J_{\alpha i}^{\beta}\partial_{v_{\beta}}$,

where we set

[TABLE]

Proof.

In this proof we make use of the following two facts. Let $f$ be a smooth function on $T^{*}M$ and let $\{\partial_{u_{i}},\partial_{v_{\alpha}},\vec{u}_{i},\vec{v}_{\alpha}\}$ be the Hamiltonian frame, one has

[TABLE]

Moreover, if $f,g$ are smooth functions on $T^{*}M$, we have $\overrightarrow{fg}=g\vec{f}+f\vec{g}$. To prove (a), one then computes

[TABLE]

where the barred terms vanishes by Fermi frame. Similarly for (b) one gets

[TABLE]

again the barred terms vanishes by Fermi frame. The expression of the coefficients $K^{\ell}_{\alpha i}$ and $J^{\beta}_{\alpha i}$ are obtained from direct computations. ∎

Lemma 12**.**

Let $v_{\alpha}(t)=\langle\lambda(t),\xi_{\alpha}|_{\gamma(t)}\rangle$, for $\alpha=1,2,3$. Then, along the geodesic, we have

[TABLE]

Proof.

Indeed $\gamma(t)=u_{i}(t)X_{i}|_{\gamma(t)}$, with $u_{i}(t)=\langle\lambda(t),X_{i}|_{\gamma(t)}\rangle$. Then, suppressing the explicit dependence on $t$ one has

[TABLE]

where the barred term vanishes along the trajectory thanks to the properties of Fermi frame. ∎

3.3. Fundamental computations

The frame $\{\partial_{u},\partial_{v},\vec{u},\vec{v}\}$ is a basis of the tangent space to $T^{*}M$. We compute the differential equations of this frame along an extremal.

Lemma 13**.**

Along the extremal $\lambda(t)$, we have

[TABLE]

where we defined the following matrices, computed along $\lambda(t)$:

[TABLE]

in the last two formulas $\{\alpha\tau\zeta\}$ is a cyclic permutation of $\{123\}$, or $\zeta=\alpha\tau$ (as product of quaternions).

Proof.

By a direct computation we get (simplifications are due to Fermi frame properties)

[TABLE]

The previous proof is completed thanks to the next lemma.

Lemma 14**.**

In terms of Biquard curvature we have

• (a)

$B_{i\ell}=-u_{j}u_{k}X_{\ell}g([X_{j},X_{i}],X_{k})=R(\dot{\gamma},X_{\ell},X_{i},\dot{\gamma})$**

• (b)

$D_{i\beta}=-u_{j}u_{k}\xi_{\beta}g([X_{j},X_{i}],X_{k})=R(\dot{\gamma},\xi_{\beta},X_{i},\dot{\gamma})$**

• (c)

$M_{\alpha\ell}=-2v_{\tau}\rho_{\zeta}(X_{l},\dot{\gamma})+\frac{1}{2}(\nabla_{X_{\ell}}T^{0})(I_{\alpha}\dot{\gamma},\dot{\gamma})$,

• (d)

$2N_{\alpha\beta}=-2v_{\tau}\rho_{\zeta}(\xi_{\beta},\dot{\gamma})+\frac{1}{2}(\nabla_{\xi_{\beta}}T^{0})(I_{\alpha}\dot{\gamma},\dot{\gamma})$,

where in the last two formulas $\{\alpha\tau\zeta\}$ is a cyclic permutation of $\{123\}$.

Proof of Lemma 14.

We use classical tricks in curvature calculations

[TABLE]

where the cancellation XX follows from properties of Fermi frame, while YY is due to the identity $u_{k}X_{k}g(\nabla_{X_{i}}X_{\ell},X_{i})=0$ (notice that the last identity is the derivative in the direction of $\dot{\gamma}(t)$ of the following one $g(\nabla_{X_{i}}X_{\ell},X_{j})|_{\gamma(t)}=0$). Thus using that the torsion among horizontal vector fields is vertical

[TABLE]

On the other hand, the term $g(\nabla_{X_{\ell}}X_{k},X_{j})$ is skew-symmetric in $k,j$ and we have a symmetric sum so the first term is zero and

[TABLE]

Along the same path one can show that

[TABLE]

Applying (41), (42), we have

[TABLE]

where $\{\beta\tau\zeta\}$ is a cyclic permutation of $\{123\}$. Further, we have using (11)

[TABLE]

where $\{\alpha\tau\zeta\}$ is a cyclic permutation of $\{123\}$, the skew-symmetric parts in $i$ and $k$ of the first line are cancelled and the first term of the second line is evaluated as follows

[TABLE]

Similarly one gets the formula for $N_{\alpha\beta}$ ∎

4. Symplectic products and canonical frame

In this section, for $n$-tuples $v,w$ of vectors, the symbol $\sigma(v,w)$ denotes the matrix

[TABLE]

whose entries are symplectic products. Notice that with this convention one has the identities

[TABLE]

where, for $n$-tuple of vectors $v$ and a matrix $L$, the juxtaposition $Lv$ denotes the $n$-tuple of vectors obtained by matrix multiplication. When $v,w$ are vector fields defined along an extremal $\lambda(t)$, the following Leibniz rule holds

[TABLE]

4.1. Derivatives of $\partial_{v}$ and symplectic products of the coordinate frame

By a direct computation one gets from Lemma 13 the following relations.

Lemma 15**.**

Along the extremal, we have

[TABLE]

To prove the fourth equality we used (13) and (15) below. We then compute symplectic products of the elements of the basis.

Lemma 16**.**

The non-zero brackets between $\partial_{u_{i}},\partial_{v_{\alpha}},\vec{u}_{i},\vec{v}_{\alpha}$ are

• (a)

$\sigma(\partial_{u},\vec{u})=\mathbbold{1}$,

• (b)

$\sigma(\partial_{v},\vec{v})=\mathbbold{1}$,

• (c)

$\sigma(\vec{u},\vec{u})=\{u_{i},u_{j}\}=-2C$,

• (d)

$\sigma(\vec{v},\vec{v})=\{v_{\alpha},v_{\beta}\}=SV+\chi$,

• (e)

$\sigma(\vec{u},\vec{v})=\{u_{i},v_{\alpha}\}=P$,

where, according to (38),

[TABLE]

and, thanks to (36),

[TABLE]

Notice that, by bilinearity of $\sigma$, Lemma 16 permits to compute the symplectic product of any pair of vectors.

Lemma 17** (Several identities).**

We have the following identities

[TABLE]

Proof.

The identities (13) follow directly from the definitions while (14) is precisely (12) written in terms of $A,C$ and $v$. For (15), working in the Fermi frame along $\gamma$, we have

[TABLE]

where we used (12) to get the last equality. ∎

Corollary 18**.**

We also have

[TABLE]

Proof.

The identities (16) follow directly from (13) and (15). For the first one in (17), we take the derivative of (15) applying (13) and (14) to get that

[TABLE]

The second identity in (17) follows from the first one applying (13). ∎

Lemma 19** (Derivative of $V$).**

We have $\dot{V}\not=0$. In particular

[TABLE]

In vector notation $\dot{v}=-G^{*}\dot{\gamma}$. In particular if $T^{0}=0$ then $\dot{v}=\dot{V}=0.$

Proof.

It easily follows by

[TABLE]

4.2. Computation of symplectic products

Now we deduce symplectic products of derivatives of the vector $\partial_{v}$ using Lemma 15, Lemma 16, Lemma 17 and Corollary 18.

Lemma 20**.**

We have

[TABLE]

Proof.

The identities (19) and the first two ones in (20) follow from Lemma 15, Lemma 16 and Lemma 17. We calculate $\sigma(\dot{\partial}_{v},\dddot{\partial}_{v})=2A(-12VA-4v\dot{\gamma}^{*})^{*}=-24AA^{*}V^{*}-8A\dot{\gamma}v^{*}=24V$ which proves the third one in (20). Differentiating the second identity in (20), one gets $0=\sigma(\ddot{\partial}_{v},\ddot{\partial}_{v})+\sigma(\dot{\partial}_{v},\dddot{\partial}_{v})=\sigma(\ddot{\partial}_{v},\ddot{\partial}_{v})+24V$ which yields the third one in (20). ∎

4.3. Canonical frame

To compute $\mathcal{R}_{cc}=\sigma(\dot{F}_{c},F_{c})$, we need to compute the elements of the canonical basis up to $F_{c}$. The algorithm to recover them (following the general construction developed in [44]) starts from identifying $E_{a}$ and then works as follows:

[TABLE]

The triplet $E_{a}$ is determined by the following four conditions:

• (i)

$\pi_{*}E_{a}=0$,

• (ii)

$\pi_{*}\dot{E}_{a}=0$,

• (iii)

$\sigma(\ddot{E}_{a},\dot{E}_{a})=\mathbbold{1}$,

• (iv)

$\sigma(\ddot{E}_{a},\ddot{E}_{a})=\mathbbold{0}$.

Items (i) and (ii) imply that there exists $M\in\mathrm{GL}(3)$ such that $E_{a}=M\partial_{v}$. Condition (iii) implies that $M=\tfrac{1}{2}O$ with $O\in\mathrm{O}(3)$. Finally, (iv) implies that $O$ satisfies the differential equation

[TABLE]

Its solution is unique up to an orthogonal transformation (the initial condition, that we set $O(0)=\mathbbold{1}$). Using the structural equations together with (21), we have

[TABLE]

Thus we can also compute

[TABLE]

The next step is to compute $E_{c}$. It is determined by the following conditions:

• (i)

$\pi_{*}E_{c}=0$,

• (ii)

$\sigma(E_{c},F_{c})=\mathbbold{1}$ and $\sigma(E_{c},F_{b})=\sigma(E_{c},F_{a})=\mathbbold{0}$,

• (iii)

$\pi_{*}\ddot{E}_{c}=0$.

For (i) we can write

[TABLE]

where $Y$ is a $(4n-3)\times 4n$ matrix and $W$ is a $(4n-3)\times 3$ matrix.

To compute $\sigma(E_{c},F_{c}),\sigma(E_{c},F_{b}),\sigma(E_{c},\dot{F}_{b})$ using (24), (22) and (23), we need to know

[TABLE]

The only non-zero terms using Lemma 15 are given by:

[TABLE]

For (ii), observing that $\sigma(E_{c},F_{a})=\mathbbold{0}$ implies $\sigma(E_{c},\dot{F}_{b})=\mathbbold{0}$, we get

[TABLE]

We obtain from (24) and (25)

[TABLE]

Finally, using (iii) and the equality above, we get that $Y$ must satisfy

[TABLE]

Using Lemma 17, (25), (26) and (18), we have

[TABLE]

Observe that $Y$ represents an orthogonal projection on $\mathcal{D}\cap\operatorname{\mathrm{span}}\{I_{\alpha}\dot{\gamma},I_{\beta}\dot{\gamma},I_{\tau}\dot{\gamma}\}^{\perp}$. Then

[TABLE]

Substitute (26) into the second equality of (24) to get

[TABLE]

We calculate from (28) using Lemma 13, (25), (26) and (27) that

[TABLE]

where the cancellations in the fourth line follow from (26) and (25).

Applying Lemma 16, Lemma 17, (25) and (26) we obtain from (28) and (29)

[TABLE]

where we used $C_{ij}C_{jk}=-v_{\alpha}v_{\beta}g(I_{\alpha}X_{i},I_{\beta}X_{k})=-||v||^{2}\mathbbold{1}$.

Lemma 21**.**

(Derivative of $C$ along $\gamma(t)$) We have

[TABLE]

Proof.

By a direct computation

[TABLE]

Finally we derive the following formula for the trace of the matrix $\mathcal{R}_{cc}$.

Lemma 22**.**

We have

[TABLE]

Proof.

Indeed, we obtain from (30)

[TABLE]

where we used $\mathrm{tr}(\dot{C}\mathbbold{1})=0$ and

[TABLE]

5. Proof of Theorems 1 and 2

Theorem 1 is a direct consequence of Theorem 7 and Lemma 22. To prove Theorem 2 we use the following Lemma to rewrite the condition in Theorem 1.

Lemma 23**.**

We have the identity

[TABLE]

Proof.

Fix a unit vector $X$ and set $Y=Z=I_{1}X$, $V=X$ into (40). One obtains the following expression

[TABLE]

Do the same for $I_{2}$, $I_{3}$ and sum the obtained equalities, we obtain applying the first Bianchi identity for the Biquard connection (39).

[TABLE]

From (39) we have $b(I_{1}X,I_{2}X,I_{3}X,X)=2T^{0}(X,X)-6U(X,X)$. Hence,

[TABLE]

and the first identity of (38) combined with (32) gives (31). ∎

Now, Theorem 2 follows from Theorem 1 and Lemma 23.

Appendix A Some technical facts and useful identities

Here we recall some properties of the torsion and curvature of the Biquard connection. See also [17, 31, 33, 28, 32] for a comprehensive exposition.

A.1. Invariant decompositions

Any endomorphism $\Psi$ of $\mathcal{D}$ can be decomposed with respect to the quaternionic structure $(\mathbb{Q},g)$ uniquely into four $\mathrm{Sp}(n)$-invariant parts $\Psi=\Psi^{+++}+\Psi^{+--}+\Psi^{-+-}+\Psi^{--+},$ where $\Psi^{+++}$ commutes with all three $I_{i}$, $\Psi^{+--}$ commutes with $I_{1}$ and anti-commutes with the others two, etc. The two $\mathrm{Sp}(n)\mathrm{Sp}(1)$-invariant components are given by

[TABLE]

They are the projections on the eigenspaces of the Casimir operator $\Upsilon=\ I_{1}\otimes I_{1}\ +\ I_{2}\otimes I_{2}\ +\ I_{3}\otimes I_{3},$ corresponding, respectively, to the eigenvalues $3$ and $-1$, see [18]. Note here that each of the three 2-forms $\omega_{s}$ belongs to the $[-1]$-component and constitute a basis of the Lie algebra $\mathrm{sp}(1)$.

If $n=1$ then the space of symmetric endomorphisms commuting with all $I_{\alpha}$ is 1-dimensional, i.e., the $[3]$-component of any symmetric endomorphism $\Psi$ on $\mathcal{D}$ is proportional to the identity, $\Psi_{[3]}=-\frac{tr\Psi}{4}\mathrm{id}_{|H}$.

A.2. The torsion tensor

The torsion endomorphism $T_{\xi}=T(\xi,\cdot):H\rightarrow H,\quad\xi\in V$ will be decomposed into its symmetric part $T_{\xi}^{0}$ and skew-symmetric part $b_{\xi},T_{\xi}=T_{\xi}^{0}+b_{\xi}$. Biquard showed in [17] that the torsion $T_{\xi}$ is completely trace-free, $tr\,T_{\xi}=tr\,T_{\xi}\circ I_{{\alpha}}=0$, its symmetric part has the properties

[TABLE]

The skew-symmetric part can be represented as $b_{\xi_{{\alpha}}}=I_{{\alpha}}U$, where $U$ is a traceless symmetric (1,1)-tensor on $\mathcal{D}$ which commutes with $I_{1},I_{2},I_{3}$. Therefore we have $T_{\xi_{{\alpha}}}=T_{\xi_{{\alpha}}}^{0}+I_{{\alpha}}U$. When $n=1$ the tensor $U$ vanishes identically, $U=0$, and the torsion is a symmetric tensor, $T_{\xi}=T_{\xi}^{0}$. The two $\mathrm{Sp}(n)\mathrm{Sp}(1)$-invariant trace-free symmetric 2-tensors on $\mathcal{D}$

[TABLE]

were introduced in [31] and enjoy the properties

[TABLE]

From [33, Proposition 2.3] we have

[TABLE]

hence, taking into account (35) it follows

[TABLE]

Any 3-Sasakian manifold has zero torsion endomorphism, and the converse is true if in addition the qc-Einstein curvature (see (37)) is a positive constant [31].

A.3. Torsion and curvature

Let $R=[\nabla,\nabla]-\nabla_{[\ ,\ ]}$ be the curvature tensor of $\nabla$ and the dimension is $4n+3$. We denote the curvature tensor of type (0,4) and the torsion tensor of type (0,3) by the same letter, $R(A,B,C,D):=g(R(A,B)C,D),\quad T(A,B,C):=g(T(A,B),C)$, $A,B,C,D\in\Gamma(TM)$. The qc-Ricci tensor $Ric$, normalized qc-scalar curvature $S$, qc-Ricci forms $\rho_{\alpha}$ of the Biquard connection are defined, respectively, by the following formulas

[TABLE]

where $X_{1},\dots,X_{4n}$ is an orthonormal basis of $\mathcal{D}$.

A qc structure is said to be qc-Einstein if the horizontal qc-Ricci tensor is a scalar multiple of the metric, $\mathrm{Ric}(X,Y)=2(n+2)Sg(X,Y).$

As shown in [31, 28] the qc-Einstein condition is equivalent to the vanishing of the torsion endomorphism of the Biquard connection. In this case $S$ is constant and the vertical distribution is integrable. It is also worth recalling that the horizontal qc-Ricci tensors and the integrability of the vertical distribution can be expressed in terms of the torsion of the Biquard connection [31] (see also [29, 33, 32]). For example, we have

[TABLE]

Note that for $n=1$ the above formulas hold with $U=0$.

A.4. Bianchi identity

We shall also need the first Bianchi identity and the general formula for the curvature [31, 33, 34].

The first Bianchi identity for the Biquard connection reads

[TABLE]

where $\sum_{(X,Y,Z)}$ denotes the cyclic sum over $\{X,Y,Z\}$ and we used that $(\nabla_{X}T)(Y,Z,V)=0$ for horizontal vectors.

We also have the identities, cf. [33, Theorem 3.1] or [34, Theorem 4.3.11],

[TABLE]

[TABLE]

where the Ricci two forms are given by

[TABLE]

A.5. Fermi frame

Here we prove the existence of Fermi frame. We recall the statement

Lemma 24**.**

Given a geodesic $\gamma(t)$, there exists a $\mathbb{Q}$-orthonormal frame, i.e., a horizontal frame $X_{i}$, $i\in\{1,\dots,4n\}$, and vertical frame $\xi_{\alpha}$, $\alpha=1,2,3$ in a neighborhood of $\gamma(0)$, such that for all $\alpha,\beta\in\{1,2,3\}$ and $i,j\in\{1,\dots,4n\}$,

• (i)

the frame is orthonormal for the Riemannian metric $g+\sum_{\beta}\eta_{\beta}^{2}$,

• (ii)

$\nabla_{X_{i}}X_{j}|_{\gamma(t)}=\nabla_{\xi_{\alpha}}X_{j}|_{\gamma(t)}=\nabla_{X_{i}}\xi_{\beta}|_{\gamma(t)}=\nabla_{\xi_{\alpha}}\xi_{\beta}|_{\gamma(t)}=0$.

In particular, for all $\alpha,\beta,\tau\in\{1,2,3\}$ and $i,j\in\{1,\dots,4n\}$

[TABLE]

Proof.

Since $\nabla$ preserves the splitting $H\oplus V$ we can apply the standard arguments for the existence of a Fermi normal frame along a smooth curve with respect to a metric connection (see e.g., [26]). We sketch the proof for completeness.

Let $\{\tilde{X}_{1},\dots,\tilde{X}_{4n},\tilde{\xi}_{1},\tilde{\xi}_{2},\tilde{\xi}_{3}\}$ be a $\mathbb{Q}$-orthonormal basis around $p=\gamma(0)$ such that $\tilde{X}_{a_{|}p}=X_{a}(p)$ and $\tilde{\xi}_{{\alpha}_{|}p}=\xi_{\alpha}(p)$. We want to find a modified frame $X_{a}=o^{b}_{a}\tilde{X}_{b}$ and $\xi_{\alpha}=o^{\tau}_{\alpha}\tilde{\xi}_{\tau},$ which satisfies the normality conditions along the smooth geodesic $\gamma(t)$.

Let $\varpi$ be the $\mathrm{sp}(n)\oplus\mathrm{sp}(1)$-valued connection 1-forms with respect to the frame $\tilde{X}_{1},\dots,\tilde{X}_{4n}$, and $\tilde{\xi}_{1},\tilde{\xi}_{2},\tilde{\xi}_{3}$, namely

[TABLE]

Consequently, we have

[TABLE]

[TABLE]

Since the Biquard connection preserves the splitting $H\oplus V$, the existence of a Fermi normal frame along $\gamma(t)$ is equivalent to the existence of a smooth solution to the system

[TABLE]

A smooth solution to this system on a small neighborhood along $\gamma(t)$ exists, see e.g., [26, Theorem 3.1]. Clearly, the solution $o^{a}_{b}$ belongs to $\mathrm{Sp}(n),o^{a}_{b}\in\mathrm{Sp}(n)$ since the connection 1-forms belong to the Lie algebra $\mathrm{sp}(n)$ and the solution $o^{\beta}_{\alpha}\in\mathrm{Sp}(1)$ because the connection 1-forms are in the Lie algebra $\mathrm{sp}(1)$. ∎

Acknowledgements

The first author has been supported by the ANR project SRGI “Sub-Riemannian Geometry and Interactions”, contract number ANR-15-CE40-0018. The second author has been partially supported by Contract DFNI I02/4/12.12.2014 and Contract 195/2016 with the Sofia University ”St.Kl.Ohridski”.