Equivariant CR minimal immersions from $S^3$ into $\mathbb{C}P^n$

TL;DR

This paper extends the classification of equivariant CR minimal immersions from the 3-sphere into complex projective spaces by removing the constant curvature assumption, using Lie algebra structure constants.

Contribution

It provides a broader classification theorem for equivariant CR minimal immersions without assuming constant sectional curvature.

Findings

Extended classification of immersions without constant curvature assumption

Utilized Lie algebra structure constants to analyze geometric properties

Established conditions for equivariant CR minimal immersions

Abstract

The equivariant CR minimal immersions from the round $3$-sphere $S^3$ into the complex projective space $\mathbb CP^n$ have been classified by the third author explicitly (J London Math Soc 68: 223-240, 2003). In this paper, by employing the equivariant condition which implies that the induced metric is left-invariant, and that all geometric properties of $S^3={\rm SU}(2)$ endowed with a left-invariant metric can be expressed in terms of the structure constants of the Lie algebra $\mathfrak{su}(2)$, we establish an extended classification theorem for equivariant CR minimal immersions from the $3$-sphere $S^3$ into $\mathbb CP^n$ without the assumption of constant sectional curvatures.

Equations

$$\phi (z,w)=[\cos \frac{\pi }{8}(z^2,\sqrt{2}zw,w^2)+\sqrt{-1}\sin \frac{\pi }{8}({\bar{w}}^2,-\sqrt{2}\bar{z}\bar{w},{\bar{z}}^2)].$$

$$\varphi(z,w)=\big{[}\cos t\sum_{\alpha=0}^{k}\sqrt{C_{\alpha}^{k}}z^{k-\alpha}w^{\alpha}\varepsilon_{\alpha}+\sqrt{-1}\sin t\sum_{\alpha^{\prime}=0}^{\ell}\sqrt{C_{\alpha^{\prime}}^{\ell}}z^{\ell-\alpha^{\prime}}w^{\alpha^{\prime}}\varepsilon^{\prime}_{\alpha^{\prime}}\big{]},$$

$$\mathrm{id}:S^3\to SU(2):(z,w)\mapsto \left(\begin{array}{cc}z& -\bar{w}\\ w& \bar{z}\end{array}\right).$$

$$\left(\begin{array}{cc}i{\omega }_1^{\prime }& -{\omega }_2^{\prime }+i{\omega }_3^{\prime }\\ {\omega }_2^{\prime }+i{\omega }_3^{\prime }& -i{\omega }_1^{\prime }\end{array}\right)=\left(\begin{array}{cc}\bar{z}& \bar{w}\\ -w& z\end{array}\right)\left(\begin{array}{cc}dz& -d\bar{w}\\ dw& d\bar{z}\end{array}\right),$$

$$d{\omega }_1^{\prime }=2{\omega }_2^{\prime }\wedge {\omega }_3^{\prime },d{\omega }_2^{\prime }=2{\omega }_3^{\prime }\wedge {\omega }_1^{\prime },d{\omega }_3^{\prime }=2{\omega }_1^{\prime }\wedge {\omega }_2^{\prime }.$$

$$\{\begin{array}{rl}{\omega }_1^{\prime }& =-\frac{i}{2}(\bar{z}dz+\bar{w}dw-zd\bar{z}-wd\bar{w}),\\ {\omega }_2^{\prime }& =-\frac{1}{2}(wdz-zdw+\bar{w}d\bar{z}-\bar{z}d\bar{w}),\\ {\omega }_3^{\prime }& =\frac{i}{2}(wdz-zdw-\bar{w}d\bar{z}+\bar{z}d\bar{w}).\end{array}$$

$$X_1^{\prime }(z,w)=i(z,w),X_2^{\prime }(z,w)=(-\bar{w},\bar{z}),X_3^{\prime }(z,w)=i(-\bar{w},\bar{z}).$$

$${\mathrm{Ad}}^{\ast }:SU(2)\times \mathfrak{su}(2{)}^{\ast }\to \mathfrak{su}(2{)}^{\ast }:(\varrho ,\theta )\mapsto {\mathrm{Ad}}_{\varrho }^{\ast }(\theta )=a_{\varrho }^{\ast }\theta ,$$

$$d{\omega }_i=-\sum {\omega }_{ij}\wedge {\omega }_j,{\omega }_{ij}+{\omega }_{ji}=0,$$

$$d{\omega }_{ij}=-\sum {\omega }_{ik}\wedge {\omega }_{kj}+{\Omega }_{ij},$$

$${\nabla }_{X_k}{X}_i=-\sum {\omega }_{ij}(X_k)X_j,{\nabla }_{X_k}{\omega }_i=-\sum {\omega }_{ij}(X_k){\omega }_j.$$

$${\omega }_{ij}=\frac{1}{2}\sum (c_{ij}^k-c_{jk}^i+c_{ik}^j){\omega }_k.$$

$$\{\begin{array}{rl}{\omega }_{12}& =c_{12}^1{\omega }_1+c_{12}^2{\omega }_2+(c_{12}^3+a){\omega }_3,\\ {\omega }_{23}& =(c_{23}^1+a){\omega }_1+c_{23}^2{\omega }_2+c_{23}^3{\omega }_3,\\ {\omega }_{31}& =c_{31}^1{\omega }_1+(c_{31}^2+a){\omega }_2+c_{31}^3{\omega }_3,\end{array}$$

$$a=-\frac{1}{2}(c_{23}^1+c_{31}^2+c_{12}^3).$$

$$d{\omega }_i=-\frac{1}{2}\sum c_{jk}^i{\omega }_j\wedge {\omega }_k=-\sum _{j<k}{c}_{jk}^i{\omega }_j\wedge {\omega }_k,$$

$$\{\begin{array}{rl}-d{\omega }_1& =c_{23}^1{\omega }_2\wedge {\omega }_3+c_{31}^1{\omega }_3\wedge {\omega }_1+c_{12}^1{\omega }_1\wedge {\omega }_2,\\ -d{\omega }_2& =c_{23}^2{\omega }_2\wedge {\omega }_3+c_{31}^2{\omega }_3\wedge {\omega }_1+c_{12}^2{\omega }_1\wedge {\omega }_2,\\ -d{\omega }_3& =c_{23}^3{\omega }_2\wedge {\omega }_3+c_{31}^3{\omega }_3\wedge {\omega }_1+c_{12}^3{\omega }_1\wedge {\omega }_2.\end{array}$$

$$C=\left(\begin{array}{ccc}{c}_{23}^1& c_{31}^1& c_{12}^1\\ c_{23}^2& c_{31}^2& c_{12}^2\\ c_{23}^3& c_{31}^3& c_{12}^3\end{array}\right)$$

$$(d{\omega }_1,d{\omega }_2,d{\omega }_3)=-\epsilon (\ast {\omega }_1,\ast {\omega }_2,\ast {\omega }_3)C.$$

$$(d{\tilde{\omega }}_1,d{\tilde{\omega }}_2,d{\tilde{\omega }}_3)=-\tilde{\epsilon }(\ast {\tilde{\omega }}_1,\ast {\tilde{\omega }}_2,\ast {\tilde{\omega }}_3)\tilde{C}.$$

$$-\tilde{\epsilon }(\ast {\omega }_1,\ast {\omega }_2,\ast {\omega }_3)T\tilde{C}=-\tilde{\epsilon }(\ast {\tilde{\omega }}_1,\ast {\tilde{\omega }}_2,\ast {\tilde{\omega }}_3)\tilde{C}=-\epsilon (\ast {\omega }_1,\ast {\omega }_2,\ast {\omega }_3)CT.$$

$$\tilde{C}=\tilde{\epsilon }\epsilon T^{-1}CT=\tilde{\epsilon }\epsilon T^{t}CT.$$

$$c_{21}^2+c_{31}^3=0,c_{12}^1+c_{32}^3=0,c_{13}^1+c_{23}^2=0.$$

$$-c_{31}^1{\omega }_1\wedge {\omega }_2\wedge {\omega }_3=d{\omega }_1\wedge {\omega }_2={\omega }_1\wedge d{\omega }_2=-c_{23}^2{\omega }_1\wedge {\omega }_2\wedge {\omega }_3.$$

$$d{\omega }_1=2a_{11}{\omega }_2^{\prime }\wedge {\omega }_3^{\prime }+2a_{21}{\omega }_3^{\prime }\wedge {\omega }_1^{\prime }+2a_{31}{\omega }_1^{\prime }\wedge {\omega }_2^{\prime }.$$

$$-c_{23}^1{\omega }_1\wedge {\omega }_2\wedge {\omega }_3={\omega }_1\wedge d{\omega }_1=2(a_{11}^2+a_{21}^2+a_{31}^2){\omega }_1^{\prime }\wedge {\omega }_2^{\prime }\wedge {\omega }_3^{\prime }.$$

$$d{s}^2=\frac{1}{c}(b^2{\omega }_1^{\prime 2}+{\omega }_2^{\prime 2}+{\omega }_3^{\prime 2})$$

$$\{{\omega }_1=\frac{b}{\sqrt{c}}{\omega }_1^{\prime },{\omega }_2=\frac{1}{\sqrt{c}}{\omega }_2^{\prime },{\omega }_3=\frac{1}{\sqrt{c}}{\omega }_3^{\prime }\}$$

$$d{\omega }_1=2b\sqrt{c}{\omega }_2\wedge {\omega }_3,d{\omega }_2=\frac{2}{b}\sqrt{c}{\omega }_3\wedge {\omega }_1,d{\omega }_3=\frac{2}{b}\sqrt{c}{\omega }_1\wedge {\omega }_2.$$

$$d{\omega }_1=2c_1{\omega }_2\wedge {\omega }_3,d{\omega }_2=2c_2{\omega }_3\wedge {\omega }_1,d{\omega }_3=2c_2{\omega }_1\wedge {\omega }_2.$$

$${\Omega }_{ij}=-\frac{1}{4}\sum [(c_{ik}^l+c_{lk}^i+c_{il}^k)(c_{jk}^m+c_{jm}^k+c_{mk}^j)+(c_{ij}^k+c_{kj}^i+c_{ik}^j)c_{lm}^k]{\omega }_l\wedge {\omega }_m.$$

Full text

2010 Mathematics Subject Classification. Primary 53C24; Secondary 53C42, 53C55.

The first two authors were supported by grants of NSFC-11371330 & 11771404, and the third author was supported by grants of NSFC-11361041.

1. Introduction

Let $\mathbb{C}P^{n}$ denote the $n$-dimensional complex projective space with almost complex structure $J$ and constant holomorphic sectional curvature $4$. It is known that there are considerable researches for the submanifolds of $\mathbb{C}P^{n}$, e.g. we would mention the study of its holomorphic, or (parallel) Lagrangian submanifolds in [4, 5, 19, 20, 21, 22], among a great many others. In particular, the interesting and important article of Bolton-Jensen-Rigoli-Woodward [2], which, by drawing upon Wolfson’s notion of a harmonic sequence and also exhibiting a number of prominent features of the related foundational work of Boruvka, Calabi, Chern, Eells, Wood and Lawson (see references as cited therein), had deeply investigated the conformal minimal immersions of the $2$-sphere ${S}^{2}$ into $\mathbb{C}P^{n}$ with constant curvature and constant Kähler angle. As a matter of fact, the paper [2] had brought enormous impact for subsequent research.

To extend the research of [2], and as a counterpart of Mashimo [17] studying the immersions from $S^{3}$ into $S^{n}$, a variety of immersions of $S^{3}$ into $\mathbb{C}P^{n}$ have been considered. For related references we refer to [3, 7, 8, 12, 13, 14, 15, 16], among which the paper of the third author [12] is of fundamental importance for us.

Following [1], an immersed submanifold $\varphi:{M}\rightarrow{\mathbb{C}P^{n}}$ is called a CR-submanifold if $TM$ can be decomposed into an orthogonal direct sum $TM=V_{1}\oplus V_{2}$ such that $JV_{1}\subset T^{\perp}M$ and $JV_{2}=V_{2}$. In this case the immersion $\varphi$ is said to be CR type.

Recall that $S^{3}=\{(z,w)\in\mathbb{C}^{2}|z\bar{z}+w\bar{w}=1\}$ can be identified with the Lie group $SU(2)$ in a standard way. By definition in [12], a mapping $\varphi:{S^{3}=SU(2)}\rightarrow\mathbb{C}P^{n}$ is said to be equivariant if $\varphi$ is compatible with the structure of Lie group $SU(2)$. For details see Sect. 3.

In [12], among many others, the third author studied equivariant CR minimal $S^{3}$ immersed in $\mathbb{C}P^{n}$ with constant sectional curvature $c$ (Theorem 6.1 in [12]). It was proved that if such an immersion $\varphi:S^{3}\rightarrow\mathbb{C}P^{n}$ is linearly full, then $n=2m^{2}-3$ for some integer $m\geq 2$ and $c=1/(m^{2}-1)$. Moreover, up to an isometry of $S^{3}$ and a holomorphic isometry of $\mathbb{C}P^{n}$, the immersion $\varphi$ is exactly the immersion defined in Example 4.4 of [12]. In [13], Li and Huang made an interesting advance by showing that any minimal CR $3$-sphere in $\mathbb{C}P^{n}$ is actually equivariant if the induced metric of $S^{3}$ has constant sectional curvature.

In this paper, following closely [12], we continue with the study of equivariant CR minimal immersion from $S^{3}$ into $\mathbb{C}P^{n}$ but without the assumption that the induced metric on $S^{3}$ has constant sectional curvatures. By employing the equivariant condition, Proposition 4.2 of [12] which implies that the induced metric is left-invariant, and that all geometric properties of $S^{3}={\rm SU}(2)$ endowed with a left-invariant metric can be expressed in terms of the structure constants of the Lie algebra $\mathfrak{su}(2)$, we can finally overcome the difficulty brought by the missing of constant sectional curvatures condition to achieve the goal of a new classification theorem. Our main results can be summarized as follows:

Theorem 1.1**.**

Let $\varphi:{S^{3}}\rightarrow{\mathbb{C}}P^{n}\ (n\geq 2)$ be a linearly full equivariant CR minimal immersion with induced metric $ds^{2}$. Then, up to an inner automorphism of ${\rm SU}(2)$ and a holomorphic isometry of $\mathbb{C}P^{n}$, the immersion $\varphi$ can be expressed as as one of the following three immersions:

1. (1)

$n=2$, and $(S^{3},ds^{2})$ is not a Berger sphere with

[TABLE] 2. (2)

$n\geq 2$, and $(S^{3},ds^{2})$ is a Berger sphere with

[TABLE]

where $t\in(0,\pi/2)$ is determined by, for nonnegative integer $k,\ell$ satisfying $k-\ell>0$ and $k+\ell+1=n$, \tan^{2}t=2k/\big{[}3(k-\ell)+\sqrt{(k+\ell)^{2}+8(k-\ell)^{2}}\big{]}, and $\{\varepsilon_{0},\ldots,\varepsilon_{k},\varepsilon_{0}^{\prime},\ldots,\varepsilon_{\ell}^{\prime}\}$ is the natural basis of $\mathbb{C}^{n+1}=\mathbb{C}^{k+1}\oplus\mathbb{C}^{\ell+1}$. In particular, if $\tan^{2}t=k/(2k-\ell+4)$, $(S^{3},ds^{2})$ has constant sectional curvature $c=1/(m^{2}-1)$ for some integer $m=(k-\ell)/2\geq 2$.

This paper is organized as follows. In Sect. 2, we consider the left-invariant metric $ds^{2}$ on $S^{3}=SU(2)$ with Lie algebra $\mathfrak{su}(2)$ and recall that the connection and curvature of $ds^{2}$ can be expressed precisely by the structure constants of $\mathfrak{su}(2)$. In particular, in terms of the structure constants, we state the criterion of $ds^{2}$ being a Berger metric (Proposition 2.2) and the criterion for a left-invariant metric $ds^{2}$ on $S^{3}$ to have constant sectional curvatures $c$ (Proposition 2.3). In Sect. 3, we introduce basic formulae for equivariant CR immersion (Lemma 3.1) and derive the classical Gauss-Weingarten formulae and Gauss-Codazzi equations. In Sect. 4, a roughly classification for all equivariant CR minimal immersions is proved. It turns out that there are only three possibilities of such immersions: a non-Berger sphere and $n=2$; a Berger sphere with $\nabla^{\bot}\xi_{0}=0$ for each $n\geq 2$; Berger spheres with $\nabla^{\bot}\xi_{0}\neq 0$ for each $n\geq 3$ (see Theorem 4.1). Here $\xi_{0}=J(\varphi_{\ast}(X_{1}))$ is a normal vector and $X_{1}$ is an unitary vector field that spans $W_{1}$. Then in Sect. 5, it is shown that all these possibilities do occur and their explicitly expressions in polynomial are presented (Propositions 5.1 and 5.2). Finally in Sect. 6, we demonstrate the uniqueness for such immersions (Theorems 6.1 and 6.2).

Acknowledgements. The authors are greatly indebted to the referee for his/her very helpful suggestions and valuable comments.

2. The Lie group $S^{3}$ and its canonical

structures

In this section, we shall review some fundamental results of $S^{3}$ which are given in [12] and will be needed in later sections. For completeness, we also give several new results which are of independent meaning.

2.1. The standard metric of $S^{3}$

Let $S^{3}=\{(z,w)\in\mathbb{C}^{2}|z\bar{z}+w\bar{w}=1\}$. It is identified with $SU(2)$ by

[TABLE]

Denote by $\mathfrak{su}(2)$ and $\mathfrak{su}(2)^{*}$ the Lie algebra consisting of all left-invariant vector fields and the vector space of all left-invariant $1$-forms on $S^{3}$, respectively. The collection of left-invariant $1$-forms $\{\omega^{\prime}_{1},\omega^{\prime}_{2},\omega^{\prime}_{3}\}$, defined by

[TABLE]

forms a global frame of $T^{*}S^{3}$. Here, and throughout this paper, we adopt the usual notation $i=\sqrt{-1}$. Exterior differentiation of (2.2) leads to the equations

[TABLE]

From (2.2) we get

[TABLE]

Thus the dual frame $\{X^{\prime}_{1},X^{\prime}_{2},X^{\prime}_{3}\}$ of $\{\omega^{\prime}_{1},\omega^{\prime}_{2},\omega^{\prime}_{3}\}$ is given by

[TABLE]

The standard metric on $S^{3}$ is defined by $ds_{0}^{2}=\sum_{i}\omega^{\prime}_{i}\otimes\omega^{\prime}_{i}$, which is bi-invariant and has constant sectional curvature $1$. Choose an orientation of $S^{3}$ such that $\omega^{\prime}_{1}\wedge\omega^{\prime}_{2}\wedge\omega^{\prime}_{3}=*1$, where $*1$ is the volume element with respect to $ds_{0}^{2}$. An orthonormal frame $\{\tilde{\omega}^{\prime}_{1},\tilde{\omega}^{\prime}_{2},\tilde{\omega}^{\prime}_{3}\}$ of $T^{*}S^{3}$ is called oriented frame if $\tilde{\omega}^{\prime}_{1}\wedge\tilde{\omega}^{\prime}_{2}\wedge\tilde{\omega}^{\prime}_{3}=*1$. If otherwise, $\tilde{\omega}^{\prime}_{1}\wedge\tilde{\omega}^{\prime}_{2}\wedge\tilde{\omega}^{\prime}_{3}=-*1$, it is called the oppositive oriented frame.

With respect to $ds_{0}^{2}$, the Hodge star operator $*:{\mathfrak{su}(2)^{*}}\rightarrow{\mathfrak{su}(2)^{*}\wedge\mathfrak{su}(2)^{*}}$ is a linear isomorphism, which maps the basis $\{\omega^{\prime}_{1},\omega^{\prime}_{2},\omega^{\prime}_{3}\}$ of $\mathfrak{su}(2)^{*}$ into a basis $\{\omega^{\prime}_{2}\wedge\omega^{\prime}_{3},\omega^{\prime}_{3}\wedge\omega^{\prime}_{1},\omega^{\prime}_{1}\wedge\omega^{\prime}_{2}\}$ of $\mathfrak{su}(2)^{*}\wedge\mathfrak{su}(2)^{*}$. Thus, by (2.3), $d=2*$ and it is also a linear isomorphism.

The following lemma can be proved easily.

Lemma 2.1**.**

Suppose $\{\tilde{\omega}^{\prime}_{1},\tilde{\omega}^{\prime}_{2},\tilde{\omega}^{\prime}_{3}\}$ is a basis of $\mathfrak{su}(2)^{*}$. Then it is an oriented frame of $T^{*}S^{3}$ (related to $ds_{0}^{2}$) if and only if it satisfies (2.3).

The adjoint representation ${\rm Ad}$ of $SU(2)$ induces an action of the Lie group $SU(2)$ on $\mathfrak{su}(2)^{*}$ by

[TABLE]

where $a_{\varrho}:SU(2)\rightarrow SU(2)$ defined by $a_{\varrho}(A)=\varrho^{-1}A\varrho$ is an inner automorphism. Since $d\circ a_{\varrho}^{*}=a_{\varrho}^{*}\circ d$, by virtue of Lemma 2.1, it is easily seen that $a_{\varrho}^{*}$ maps an oriented frame $\{\tilde{\omega}^{\prime}_{1},\tilde{\omega}^{\prime}_{2},\tilde{\omega}^{\prime}_{3}\}$ into an oriented frame $\{a_{\varrho}^{*}\tilde{\omega}^{\prime}_{1},a_{\varrho}^{*}\tilde{\omega}^{\prime}_{2},a_{\varrho}^{*}\tilde{\omega}^{\prime}_{3}\}$. Conversely, one can prove the following lemma.

Lemma 2.2**.**

For any given oriented frame $\{\tilde{\omega}^{\prime}_{1},\tilde{\omega}^{\prime}_{2},\tilde{\omega}^{\prime}_{3}\}$ of $T^{*}S^{3}$, there exists an element $\varrho\in SU(2)$ such that $a_{\varrho}^{*}\tilde{\omega}^{\prime}_{1}=\omega^{\prime}_{1}$, $a_{\varrho}^{*}\tilde{\omega}^{\prime}_{2}=\omega^{\prime}_{2}$ and $a_{\varrho}^{*}\tilde{\omega}^{\prime}_{3}=\omega^{\prime}_{3}$.

Remark 2.1*.*

Lemma 2.2 implies that $SU(2)$ acts transitively on $\mathfrak{su}(2)^{*}$ by $Ad^{*}$, and it also acts transitively on the set of all orientated frame.

2.2. The structure constants of the Lie algebra

$\mathfrak{su}(2)$

Now, we examine the properties of a general left-invariant metric on $S^{3}$. Some results in this subsection expand that of Milnor [18].

From now on, we adopt the convention of range of indices: $i,j,k,\ldots=1,2,3$.

A left-invariant metric on $S^{3}=SU(2)$ has the form that $ds^{2}=\sum\omega_{i}\otimes\omega_{i}$, where $\{\omega_{i}\}$ is a basis of $\mathfrak{su}(2)^{*}$. Obviously, $\{\omega_{i}\}$ is a global orthonormal frame of $T^{*}S^{3}$.

Related to $\{\omega_{i}\}$, the structure equations of $ds^{2}$ are given by

[TABLE]

[TABLE]

where $\omega_{ij}$ and $\Omega_{ij}$ are the connection forms and curvature forms, respectively.

Let $\{X_{i}\}$ be the dual frame of $\{\omega_{i}\}$ defined on $S^{3}$. Denote by $\nabla$ the Levi-Civita connection of $ds^{2}$. Then we have

[TABLE]

Let $\{c^{k}_{ij}\}$ be the structure constants of $\mathfrak{su}(2)$ with respect to the basis $\{X_{1},X_{2},X_{3}\}$, i.e., $[X_{i},X_{j}]=\sum c^{k}_{ij}X_{k}$. It is clear that $c^{k}_{ij}=-c^{k}_{ji}$.

Using the Koszul formula one gets

[TABLE]

Equivalently, the above relations can be written exactly as

[TABLE]

where

[TABLE]

Hence, $\omega_{ij}\in\mathfrak{su}(2)^{*}$, and therefore, by (2.7), $\Omega_{ij}\in\mathfrak{su}(2)^{*}\wedge\mathfrak{su}(2)^{*}$ and it is also left-invariant. Substituting (2.9) into (2.6), we can write

[TABLE]

i.e.,

[TABLE]

Since $d:{\mathfrak{su}(2)^{*}}\rightarrow{\mathfrak{su}(2)^{*}\wedge\mathfrak{su}(2)^{*}}$ is an isomorphism, the coefficients matrix on the right hand of (2.13):

[TABLE]

is non-degenerate, namely $\det C\neq 0$.

It is obvious that the matrix $C$ depends on the frame $\{\omega_{i}\}$. Assume $\{\tilde{\omega}_{i}\}$ is another orthonormal basis of $\mathfrak{su}(2)^{*}$, then a similar relations as (2.13) holds with corresponding matrix $\widetilde{C}$. Now, we derive the relationship between $C$ and $\widetilde{C}$.

Take an orientation for $S^{3}$. The volume element of $ds^{2}$ is still denoted by $*1$. With respect to $ds^{2}$, the Hodge star operator ${*}:{\mathfrak{su}(2)^{*}}\rightarrow{\mathfrak{su}(2)^{*}\wedge\mathfrak{su}(2)^{*}}$ is also a linear isomorphism. Let $\omega_{1}\wedge\omega_{2}\wedge\omega_{3}=\varepsilon*1$, $\varepsilon=\pm 1$. Then, by (2.13),

[TABLE]

Similarly, if $\{\tilde{\omega}_{i}\}$ is another orthonormal frame with $\tilde{\omega}_{1}\wedge\tilde{\omega}_{2}\wedge\tilde{\omega}_{3}=\tilde{\varepsilon}*1$, then

[TABLE]

Noting that there exists $T\in O(3)$ such that $(\tilde{\omega}_{1},\tilde{\omega}_{2},\tilde{\omega}_{3})=(\omega_{1},\omega_{2},\omega_{3})T$, then by the linearity of $d$ and $*$ we have

[TABLE]

It follows that

[TABLE]

We will call an orthonormal frame $\{\omega_{i}\}$ a normalized frame if it corresponds to a diagnalized matrix $C={\rm diag}\,(c_{23}^{1},c_{31}^{2},c_{12}^{3})$.

Before ending this subsection, we summarize the useful results in the following proposition.

Proposition 2.1**.**

(i)* The matrix $C$ in (2.14) is symmetric. Consequently,*

[TABLE]

(ii)* There is always a normalized frame $\{\omega_{i}\}$ for $ds^{2}$.*

(iii)* The scalar $a=-\tfrac{1}{2}{\rm trace}\,C$ defined by (2.11) is an orientation invariant. Moreover, $a^{2}$ is an intrinsic invariant of $ds^{2}$.*

(iv)* If $\{\omega_{i}\}$ is a normalized frame, then $c_{jk}^{i}c_{ij}^{k}>0$ for distinct $i,j,k$.*

Proof.

(i). Since $\omega_{1}\wedge\omega_{2}\in\mathfrak{su}(2)^{*}\wedge\mathfrak{su}(2)^{*}$ and that $d:{\mathfrak{su}(2)^{*}}\rightarrow{\mathfrak{su}(2)^{*}\wedge\mathfrak{su}(2)^{*}}$ is an isomorphism, we have $d(\omega_{1}\wedge\omega_{2})=0$ and $d\omega_{1}\wedge\omega_{2}=\omega_{1}\wedge d\omega_{2}$. Then, by (2.13),

[TABLE]

Thus, $c_{31}^{1}=c_{23}^{2}$. Similarly, we have $c_{12}^{1}=c_{23}^{3}$ and $c_{12}^{2}=c_{31}^{3}$. Hence (2.16) follows.

From (2.15) and (i), and according to the theory of linear algebra, we obtain immediately (ii) and (iii).

(iv). Let $\{\omega^{\prime}_{i}\}$ be an oriented frame of the standard metric $ds_{0}^{2}$. Since both $\{\omega_{i}\}$ and $\{\omega^{\prime}_{i}\}$ are basis of $\mathfrak{su}(2)^{*}$, there is a matrix $A=(a_{ij})\in GL(3,\mathbb{R})$ such that $(\omega_{1},\omega_{2},\omega_{3})=(\omega^{\prime}_{1},\omega^{\prime}_{2},\omega^{\prime}_{3})A$. Then $\omega_{1}\wedge\omega_{2}\wedge\omega_{3}=(\det A)\,\omega^{\prime}_{1}\wedge\omega^{\prime}_{2}\wedge\omega^{\prime}_{3}$ and, by Lemma 2.1, it holds that

[TABLE]

On the other hand, by assumption, $d\omega_{1}=-c_{23}^{1}\omega_{2}\wedge\omega_{3}$, thus

[TABLE]

This implies that $c_{23}^{1}\det A<0$. Similarly, we get $c_{31}^{2}\det A<0$ and $c_{12}^{3}\det A<0$. Then the assertion immediately follows. ∎

2.3. The Berger metric on $S^{3}$

First we state the following definition of Berger sphere (cf. [9, 23])

Definiton 2.1**.**

Let $\{\omega^{\prime}_{i}\}$ be an oriented frame of $S^{3}$ with respect to $ds^{2}_{0}$, and $b,c$ are two positive real numbers. Then, the left-invariant metric defined by

[TABLE]

is called a Berger metric. The $3$-sphere $S^{3}$ equipped with a Berger metric is called a Berger sphere.

Remark 2.2*.*

In [9, 23], a Berger metric is defined essentially in the form (2.17) with the frame $\{\omega^{\prime}_{i}\}$ being defined by (2.2). Thus, according to Lemma 2.2, up to an inner automorphism ${a_{\varrho}}:{SU(2)}\rightarrow{SU(2)}$, the above definition of Berger sphere should be the same as that in [9, 23]. Note that $b$ and $c$ are two parameters in a Berger metric $ds^{2}$, and its sectional curvatures are constant if and only if $b=1$.

Take $\omega^{\prime}_{1}\wedge\omega^{\prime}_{2}\wedge\omega^{\prime}_{3}$ as the orientation of $S^{3}$. Then, for the metric (2.17),

[TABLE]

is an orthonormal oriented frame. Using Lemma 2.1 we get

[TABLE]

Conversely, we have the following criterion for Berger sphere.

Proposition 2.2**.**

A left-invariant metric $ds^{2}$ on $S^{3}$ is a Berger metric if and only if there exists an orthonormal frame $\{\omega_{i}\}$ and constants $c_{1},c_{2}\in\mathbb{R}$ such that

[TABLE]

Proof.

The “only if” part is trivially from (2.18). To prove the “if” part, we first note by replacing $\omega_{1}$ with $-\omega_{1}$ if necessary we may assume $c_{1}>0$. Then the item (iv) of Proposition 2.1 implies that $c_{2}>0$. Let $c=c_{1}c_{2}$, $b=\sqrt{c_{1}/c_{2}}$. Then $c_{1}=b\sqrt{c}$, $c_{2}=\sqrt{c}/b$, and (2.19) becomes (2.18). Put $\omega^{\prime}_{1}=(\sqrt{c}/b)\omega_{1}$, $\omega^{\prime}_{2}=\sqrt{c}\omega_{2}$, $\omega^{\prime}_{3}=\sqrt{c}\omega_{3}$, then $\{\omega^{\prime}_{i}\}$ is a basis of $\mathfrak{su}(2)^{*}$ which satisfies (2.3). According to Lemma 2.1, $\{\omega^{\prime}_{i}\}$ is an oriented frame with respect to $ds_{0}^{2}$. Thus the metric $ds^{2}=\sum\omega_{i}^{2}$ assumes the form (2.17). ∎

2.4. The curvature of left-invariant metrics on

$S^{3}$

As continuation of the previous subsections, we now compute the curvatures of the left-invariant metric $ds^{2}$. As the main result, we give the necessary and sufficient conditions under which $ds^{2}$ is of constant sectional curvatures.

From (2.7), (2.9) and (2.12) we have

[TABLE]

Let $R_{ijlm}$ denote the components of the Riemannian curvature tensor of $ds^{2}$, thus $\Omega_{ij}=\tfrac{1}{2}\sum R_{ijlm}\omega_{l}\wedge\omega_{m}$ and $R_{ijlm}=-R_{ijml}$. We use notation $i,j,m\neq$ to indicate $i,j,m$ are distinct indices. Since $i,j,l,m\in\{1,2,3\}$, if $R_{ijlm}\neq 0$ then $l$ has to be $i$ or $j$, and $i,j,m\neq$. Making use (2.11) and (2.16), we obtain

[TABLE]

In particular, if $\{\omega_{i}\}$ is a normalized frame, then

[TABLE]

The following result, which is of independent meaning, establishes a criterion in terms of the structure constants for the left-invariant metric $ds^{2}$ to have constant sectional curvatures $c$.

Proposition 2.3**.**

A left-invariant metric $ds^{2}$ has constant curvature $c$ if and only if for any orthonormal frame $\{\omega_{i}\}$, the corresponding structure constants $\{c_{ij}^{k}\}$ satisfy

[TABLE]

Proof.

First of all, if a metric on $S^{3}$ has constant sectional curvature $c$, then by Cartan-Hadamard theorem we have $c>0$.

The “if” part is a direct consequence of (2.20), (2.22) and (2.11).

To prove the “only if” part, by virtue of Proposition 2.1, we can first choose a normalized frame $\{\tilde{\omega}_{i}\}$ and denote by $\{\tilde{c}^{k}_{ij}\}$ the corresponding structure constants. Then by (2.21) we have

[TABLE]

If $\tilde{c}_{mi}^{j},\tilde{c}_{ij}^{m},\tilde{c}_{jm}^{i}$ are mutually distinct, then by (2.23) $\tilde{c}_{mi}^{j}=\tilde{c}_{ij}^{m}=\tilde{c}_{jm}^{i}=0$, a contradiction. So, without loss of generality, we may assume that $\tilde{c}_{mi}^{j}=\tilde{c}_{ij}^{m}$. By (2.23) again we have $\tilde{c}_{mi}^{j}=\tilde{c}_{ij}^{m}=\tilde{c}_{jm}^{i}=\pm 2\sqrt{c}$. Thus, $\widetilde{C}=\pm 2\sqrt{c}I$ with $I$ denoting the unit matrix. Finally, let $\{\omega_{i}\}$ be an arbitrary orthonormal frame, then by (2.15) we have $C=\varepsilon T\widetilde{C}T^{-1}=\pm 2\varepsilon\sqrt{c}I$ with $\varepsilon=\pm 1$. The assertion is verified. ∎

Remark 2.3*.*

Proposition 2.3 implies that a left-invariant metric on $S^{3}$ is homothetic to the round one if and only if $C$ is proportional to the identity. Similarly, as indicated by Proposition 2.2, a left-invariant nonconstant sectional curvature metric on $S^{3}$ is a Berger metric if and only if the matrix $C$ has exactly two nonzero distinct eigenvalues.

3. Equivariant CR immersions from $S^{3}$ into $\mathbb{C}P^{n}$

In general, an equivariant map $\varphi:{G_{1}/H_{1}}\rightarrow{G_{2}/H_{2}}$ is such a map which is compatible with the Lie group structures of $G_{1}$ and $G_{2}$ (cf. [12]). In particular, for $S^{3}=SU(2)/SU(1)$ and $\mathbb{C}P^{n}=U(n+1)/[U(1)\times U(n)]$, a map $\varphi:S^{3}\rightarrow\mathbb{C}P^{n}$ is equivariant if there exist a Lie group homomorphism $E:{SU(2)}\rightarrow{U(n+1)}$ and a holomorphic isometry $A:\mathbb{C}P^{n}\rightarrow\mathbb{C}P^{n}$ such that $\varphi=A\circ\pi_{1}\circ E$, where $\pi_{1}:{U(n+1)}\rightarrow\mathbb{C}P^{n}$ is the natural projection (cf. Definition 4.1 in [12]).

Following [1], a submanifold $M\hookrightarrow\mathbb{C}P^{n}$ is called a CR-submanifold if there is an orthogonal direct sum decomposition $TM=V_{1}\oplus V_{2}$ such that $JV_{1}\subset T^{\perp}M$ and $JV_{2}=V_{2}$. In particular, if $V_{2}=\{0\}$ (resp. $V_{1}=\{0\}$), then $M$ is called a totally real (resp. complex) submanifold.

From now on, we will use the following range convention of indices:

[TABLE]

In sequel of this section, we will present the fundamental structure equations and several geometric conclusions for immersions from $S^{3}$ into $\mathbb{C}P^{n}$ under the equivariant and CR type conditions. This particularly includes Theorem 3.1, which states that an equivariant CR immersion $\varphi:S^{3}\rightarrow\mathbb{C}P^{n}$ can not be totally geodesic.

3.1. Basic formula for equivariant CR immersion $\varphi:S^{3}\rightarrow\mathbb{C}P^{n}$

Let $\pi:S^{2n+1}\rightarrow\mathbb{C}P^{n}:v\mapsto\pi(v)=[v]$ be the Hopf fibration. We suppose $\varphi:S^{3}\rightarrow\mathbb{C}P^{n}$ is an equivariant immersion. According to Proposition 4.2 of [12], there exists a global unitary frame $\{e_{0},e_{A}\}$ of the trivial bundle $\underline{\mathbb{C}}^{n+1}=S^{3}\times\mathbb{C}^{n+1}$ such that $\varphi=[e_{0}]$ and

[TABLE]

where $\rho_{0}\in\mathfrak{su}(2)^{*}$, $\theta_{A},\theta_{AB}\in\mathfrak{su}(2)^{*}\otimes\mathbb{C}$. Hence, the induced metric $ds^{2}=\sum\theta_{A}\bar{\theta}_{A}$ via $\varphi$ is left-invariant. Take an orthonormal frame $\{\omega_{i}\}$, which is a basis of $\mathfrak{su}(2)^{*}$, for $T^{*}S^{3}$. We can write $\theta_{A}=\sum a_{Ai}\omega_{i}$, where $a_{Ai}\in\mathbb{C}$ are all constant. Then

[TABLE]

Let $\Omega$ be the Kähler form of $\mathbb{C}P^{n}$, then we have

[TABLE]

where

[TABLE]

are also constant. The above equation together with (3.2) gives

[TABLE]

The tensor $-\varphi^{*}\Omega$ determines a well defined bundle endomorphism

[TABLE]

where $\{X_{i}\}$ is the dual frame of $\{\omega_{i}\}$. Since $J_{ij}$’s are constant, the rank of $F$ should be [math] or $2$. If $F=0$, then the submanifold $\varphi(S^{3})$ is totally real (or called weakly Lagrangian [10]). If $F\neq 0$, the tangent bundle $TS^{3}$ can be decomposed into orthogonal direct sum $TS^{3}=V_{1}\oplus V_{2}$ with $V_{1}=\ker F$. By re-choosing the frame we may assume $X_{1}\in V_{1}$. Then

[TABLE]

Denote by $g$ the standard Fubini-Study metric of $\mathbb{C}P^{n}$ with constant holomorphic sectional curvature $4$. From

[TABLE]

we see that $|J_{23}|\leq 1$. If $|J_{23}|=1$, then $\varphi(S^{3})$ is a CR-submanifold and $\varphi$ is said to be of CR type. In fact, $FX_{i}$ is just the tangent component of $J\varphi_{*}(X_{i})$ if we identify $TS^{3}$ with $\varphi_{*}(TS^{3})$.

Denote $\chi:=\arccos J_{23}$ the angle function. $\varphi(S^{3})$ is a totally real submanifold (resp. CR-submanifold) if and only if $\chi=\pi/2$ (resp. $\chi=0$ or $\pi$). Therefore, the angle function $\chi$ is an analogue of the Kähler angle of an immersed surface in $\mathbb{C}P^{n}$.

From now on, we assume that the immersion $\varphi$ is of CR type. The following lemma gives basic formulae which in sequel will be used from time to time.

Lemma 3.1**.**

Let $\varphi:S^{3}\rightarrow\mathbb{C}P^{n}$ be an equivariant CR immersion with induced metric $ds^{2}$. Then there exist an orthonormal frame $\{\omega_{i}\}$ of $T^{*}S^{3}$ and a unitary frame $\{e_{0},e_{1},e_{2},e_{\alpha}\}$ of $\underline{\mathbb{C}}^{n+1}$ such that $\varphi=[e_{0}]$, and

[TABLE]

where $\omega=\omega_{2}+i\omega_{3}$, $\rho_{0},\,\rho_{1},\,\rho_{2}\in\mathfrak{su}(2)^{*}$, $\theta_{12},\,\theta_{1\alpha},\,\theta_{2\alpha}$, $\theta_{\alpha\beta}\in\mathfrak{su}(2)^{*}\otimes\mathbb{C}$.

Proof.

By assumption $J_{23}=\pm 1$ in (3.6). This ensures us to choose a left-invariant orthonormal frame $\{X_{i}\}$ of $TS^{3}$, by replacing $X_{2}$ with $-X_{2}$ if necessary, such that

[TABLE]

On the other hand, as in (3.1) there exists a unitary fame $\{e_{0},e^{\prime}_{A}\}$ of $\underline{\mathbb{C}}^{n+1}$ such that $\varphi=[e_{0}]$ and

[TABLE]

where $\rho_{0}\in\mathfrak{su}(2)^{*}$, $\theta_{A},\theta^{\prime}_{AB}\in\mathfrak{su}(2)^{*}\otimes\mathbb{C}$.

Let $\{\omega_{i}\}$ be the dual frame of $\{X_{i}\}$. We can write $\theta_{A}=\sum a_{Ai}\omega_{i}$. Then (3.5) holds true with $J_{12}=J_{13}=0$, $J_{23}=1$. The first equation of (3.9) becomes

[TABLE]

where $f_{i}=\sum a_{Ai}e^{\prime}_{A}$.

Denote by $\langle\cdot,\cdot\rangle$ the canonical symmetric scalar product of $\mathbb{C}^{n+1}$. From (3.5) and (3.8) we get

[TABLE]

This leads to $|f_{2}+if_{3}|^{2}=0$ and therefore $f_{3}=if_{2}$. Thus, by (3.10),

[TABLE]

where $\omega=\omega_{2}+i\omega_{3}$ and

[TABLE]

By (3.5) and (3.8), we see that $v_{1}=(a_{11},\ldots,a_{n1})$ and $v_{2}=(a_{12},\ldots,a_{n2})$ are two orthogonal unit vectors of $\mathbb{C}^{n}$. Expand $\{v_{1},v_{2}\}$ to be a unitary basis $\{v_{1},v_{2},v_{3},\ldots,v_{n}\}$ of $\mathbb{C}^{n}$ such that $v_{\alpha}=(v_{1\alpha},\ldots,v_{n\alpha}),\ \alpha=3,\ldots,n$. Put

[TABLE]

Then $\{e_{0},e_{A}\}$ forms a unitary frame of $\underline{\mathbb{C}}^{n+1}$. According to (3.11), the structure equations of the frame $\{e_{0},e_{1},e_{2},e_{\alpha}\}$ take the form of (3.7). From (3.9), (3.12) and (3.13), we know that $\theta_{AB}=\langle{de_{A}},{\bar{e}_{B}}\rangle\in\mathfrak{su}(2)^{*}\otimes\mathbb{C}$ since $\theta^{\prime}_{AB}\in\mathfrak{su}(2)^{*}\otimes\mathbb{C}$. ∎

3.2. Invariants of equivariant CR immersion

Hereafter, $\{X_{i}\}$ is assumed the orthonormal frame satisfying (3.8) and $\{\omega_{i}\}$ is its dual frame. Set $Z=\frac{1}{2}(X_{2}-iX_{3})$ and $\omega=\omega_{2}+i\omega_{3}$. Then $\{X_{1},Z,\bar{Z}\}$ and $\{\omega_{1},\omega,\bar{\omega}\}$ give a new pair of dual frames. Obviously,

[TABLE]

From (2.8), with respect to $\{X_{1},Z,\bar{Z}\}$ and $\{\omega_{1},\omega,\bar{\omega}\}$, we have

[TABLE]

where $\sigma=\omega_{12}+i\omega_{13}$, and by (2.10),

[TABLE]

where

[TABLE]

Then, by (3.15) and (3.16), we have

[TABLE]

The structure equation (2.6), (2.7) can be rewritten as

[TABLE]

[TABLE]

With respect to $\{\omega_{1},\omega,\bar{\omega}\}$, the curvature forms can be expressed as

[TABLE]

It follows that

[TABLE]

[TABLE]

For any orthonormal frame $\{\tilde{X}_{i}\}$ with $F\tilde{X}_{1}=0$ and $F\tilde{X}_{2}=\tilde{X}_{3}$ we must have

[TABLE]

where $t$ is a real number, and $T$ is the transition matrix from $\{X_{i}\}$ to $\{\tilde{X}_{i}\}$. If we set $Z^{\prime}=\tfrac{1}{2}(\tilde{X}_{2}-i\tilde{X}_{3})$, then the above frame transformation can be written as

[TABLE]

where $\varepsilon=\det T$.

Now we prove the following proposition.

Proposition 3.1**.**

Let $\varphi:S^{3}\rightarrow\mathbb{C}P^{n}$ be an equivariant CR immersion with induced metric $ds^{2}$. Then we have:

(i)* $c_{12}^{1}=c_{13}^{1}=0$ if and only if $\nabla_{X_{1}}X_{1}=0$, that is, the integral curve of $X_{1}$ is a geodesic of $(S^{3},ds^{2})$;*

(ii)* $c_{12}^{1}=c_{13}^{1}=c_{21}^{2}=0$ and $c_{31}^{2}=c_{12}^{3}$ if and only if $(S^{3},ds^{2})$ is a Berger sphere;*

(iii)* Under the frame transformation (3.23), the structure constants $|c_{12}^{1}+ic_{13}^{1}|$, $|c_{23}^{1}|$ and $|c_{21}^{2}+\tfrac{i}{2}(c_{31}^{2}-c_{12}^{3})|$ are all invariant.*

Proof.

The assertion (i) is from the first equation of (3.18). The assertion (ii) is a direct consequence of Proposition 2.2, (3.17) and (3.19). Finally, we obtain from (3.18) that

[TABLE]

The assertion (iii) follows from the above computation of $c_{23}^{1},\,\mu$ and $\tau$. ∎

3.3. The Gauss-Weingarten formulae

Let $\widetilde{\mathbb{C}}^{n+1}=\mathbb{C}P^{n}\times\mathbb{C}^{n+1}$ be the trivial bundle, $L$ the canonical line bundle over $\mathbb{C}P^{n}$. Denote by $L^{\bot}$ (resp. $\bar{L}$) the orthogonal complement (resp. the complex conjugate) of $L$ in $\widetilde{\mathbb{C}}^{n+1}$. The trivial connection $d$ of $\widetilde{\mathbb{C}}^{n+1}$ induces the standard connection on both $L$ and $L^{\bot}$. This produces the connection on $\bar{L}$ and hence on $\bar{L}\otimes L^{\bot}$.

To clarify the above let us take a local unitary frame $\{e_{0},\,e_{A}\}$ of $\widetilde{\mathbb{C}}^{n+1}$ with $e_{0}\in L$. For simplicity, we will denote by the same notation $\nabla$ the connections on different vector bundles. Let us write

[TABLE]

Then by definition

[TABLE]

Thus $\nabla\bar{e}_{0}=-i\rho_{0}\bar{e}_{0}$, and consequently,

[TABLE]

If we identify $T\mathbb{C}P^{n}\cong_{\mathbb{R}}T^{(1,0)}\mathbb{C}P^{n}$ with $\bar{L}\otimes L^{\bot}$, then the connection on $\bar{L}\otimes L^{\bot}$ is exactly the Levi-Civita connection on $T\mathbb{C}P^{n}$. With respect to the unitary frame $\{\bar{e}_{0}\otimes e_{A}\}$ of $\bar{L}\otimes L^{\bot}$, $\{\theta_{AB}-i\delta_{AB}\rho_{0}\}$ are the connection $1$-forms.

For an immersion $\varphi:{S^{3}}\rightarrow{\mathbb{C}P^{n}}$, the pullback bundle of $\widetilde{\mathbb{C}}^{n+1}$ via $\varphi$ will be denoted also by $\underline{\mathbb{C}}^{n+1}$, whereas those of $L$, $\bar{L}$ and $L^{\bot}$ via $\varphi$ will still be denoted by the same notations, respectively. Using Lemma 3.1 we have a global unitary frame $\{e_{0},e_{A}\}$ of $\underline{\mathbb{C}}^{n+1}$ with $\varphi=[e_{0}]$ such that (3.7) holds. Then the differential (tangent map) of $\varphi$, namely that

[TABLE]

is given by $d\varphi=\omega_{1}(\bar{e}_{0}\otimes e_{1})+\omega(\bar{e}_{0}\otimes e_{2})$, or equivalently,

[TABLE]

Then, the tangent bundle and normal bundle of $\varphi(S^{3})$ are given, respectively, by

[TABLE]

Denote by $\bar{\nabla}$ the connection on the pullback bundle $\varphi^{-1}T\mathbb{C}P^{n}$. From (3.24) and (3.25), together with (3.7), we see that

[TABLE]

Denote by $B$ the second fundamental form of $\varphi$. Then the Gauss formula

[TABLE]

implies that the tangential component of the right hand side of (3.27) is $\varphi_{*}(\nabla X_{j})$.

From (3.15) and (3.25), we get

[TABLE]

It follows that

[TABLE]

Similarly, the above fact, together with (3.25) and $\nabla X_{2}=\omega_{12}X_{1}-\omega_{23}X_{3}$, gives

[TABLE]

Thus we get

[TABLE]

Note that the second fundamental form can be expressed by

[TABLE]

By using (3.27) and (3.28), we have

[TABLE]

Since, by (3.16), $\sigma=\omega_{12}+i\omega_{13}\neq 0$, and that $\det C\not=0$ implies that $\{c_{ij}^{1}\}$ can not be all zero, it follows that $B\neq 0$. Hence we have

Theorem 3.1**.**

An equivariant CR immersion $\varphi:S^{3}\rightarrow\mathbb{C}P^{n}$ can not be totally geodesic.

From (3.26) we see that the set of normal vector fields:

[TABLE]

defines an orthonormal frame of $T^{\bot}S^{3}$. Moreover, by (3.24), we have

[TABLE]

Then, according to the Weingarten formula

[TABLE]

we get

[TABLE]

[TABLE]

Put $W_{1}:={\rm span}\{\xi_{0}\}$ and $W_{2}:=W_{1}^{\perp}$ in $T^{\bot}S^{3}$. If $A_{\xi}=0$ for any $\xi\in W_{2}$, the submanifold $\varphi(S^{3})$ is said to be totally geodesic with respect to $W_{2}$. Then, from the above calculations, we have the following observations:

Proposition 3.2**.**

(i)* $\xi_{0}$ is parallel in $T^{\bot}S^{3}$ $($that is $\nabla^{\perp}\xi_{0}=0)$ if and only if $\theta_{1\alpha}=0$. (ii) The submanifold $\varphi(S^{3})$ is totally geodesic with respect to $W_{2}$ if and only if $\theta_{1\alpha}=\theta_{2\alpha}=0$.*

3.4. The Gauss-Codazzi equations

Exterior differentiation of (3.7), and using (3.28), leads to the following eqs.:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

From (3.30) we see that the symmetry of $B$ is given by (3.34). In fact, by using (2.6), the first equation of (3.34) means that

[TABLE]

Substituting (2.10) into the above equation, and using (2.16), we get

[TABLE]

The exterior differentiation of (3.29), together with (3.33), (3.36), (3.37) and (3.20), gives the Gauss equation

[TABLE]

Similarly, from (3.27), noting that $\sigma=-\theta_{12}$, we see that (3.37) and (3.38) are part of the Codazzi equations, whereas the rest one is

[TABLE]

4. Equivariant CR minimal immersions from $S^{3}$ into $\mathbb{C}P^{n}$

The purpose of this section is the proof of Theorem 4.1, which concerns with a roughly classification of equivariant CR minimal immersions from $S^{3}$ into $\mathbb{C}P^{n}$. Recall that a equivariant CR immersion $\varphi:S^{3}\rightarrow\mathbb{C}P^{n}$ is minimal if and only if ${\rm trace}\,B=0$, that is, by (3.30),

[TABLE]

This, by the use of (2.10) and (2.11), is equivalent to

[TABLE]

The above two equations, together with (3.40), (2.10) and the second equation of (3.34), lead to

[TABLE]

[TABLE]

where $\lambda_{\alpha},\;\mu_{\alpha}\in\mathbb{C}$ are constant.

Recall that a submanifold $M\hookrightarrow\mathbb{C}P^{n}$ is called linearly full if there is no totally geodesic $\mathbb{C}P^{k}\;(k<n)$ such that $M\hookrightarrow\mathbb{C}P^{k}\subset\mathbb{C}P^{n}$. For such a submanifold the dimension $n$ is called the full dimension. Henceforth, we assume that $\varphi:S^{3}\rightarrow\mathbb{C}P^{n}$ is an equivariant CR minimal immersion with full dimension $n$.

First of all, as the continuation of Proposition 3.2 (ii), we have

Lemma 4.1**.**

(i)* If $\varphi:S^{3}\rightarrow\mathbb{C}P^{n}$ is an equivariant CR immersion with the full dimension $n$ and that $\varphi(S^{3})$ is totally geodesic with respect to $W_{2}$, then $n=2$.*

(ii)* If $\varphi:S^{3}\to\mathbb{C}P^{2}$ is an equivariant CR minimal immersion, then the structure constants $\{c_{ij}^{k}\}$ with respect to an orthonormal basis $\{X_{1},X_{2},X_{3}\}$ and coefficients forms of equation (3.7) satisfy:*

[TABLE]

[TABLE]

Proof.

(i) If $\varphi(S^{3})$ is totally geodesic with respect to $W_{2}$, then, from Proposition 3.2, we have $\theta_{1\alpha}=\theta_{2\alpha}=0$. Since $\theta_{12}=-\sigma$, the first three equations in (3.7) become

[TABLE]

Then by (3.33) and (3.36) we get $d(\rho_{0}+\rho_{1}+\rho_{2})=0$. It follows by the isomorphism of $d:\mathfrak{su}(2)^{*}\to{\mathfrak{su}(2)^{*}\wedge\mathfrak{su}(2)^{*}}$ that $\rho_{0}+\rho_{1}+\rho_{2}=0$. Then we get $d(e_{0}\wedge e_{1}\wedge e_{2})=0$, showing that ${\it span}\{e_{0},e_{1},e_{2}\}$ is a constant subspace. The fullness condition thus implies that $n=2$.

(ii) In this situation, as above we still have $\rho_{0}+\rho_{1}+\rho_{2}=0$. Thus, by (3.29) and (4.2), it holds

[TABLE]

Taking exterior differentiation of (4.7), together with the use of (3.33) and the first equation of (3.19), gives

[TABLE]

Then, by (3.16), we have

[TABLE]

Since $\mu=0$, from (3.17) and (2.16), we see that $c_{23}^{2}=c_{23}^{3}=0$. Then, by (4.2), (3.29) and that $\rho_{0}=(a/3)\omega_{1}$, we get

[TABLE]

This completes the proof of (4.4).

Next, by (3.19), $d\omega_{1}=-\tfrac{i}{2}c_{23}^{1}\omega\wedge\bar{\omega}$. Then, using $ac_{23}^{1}=-6$, (4.10), (3.36) and (4.9), we get

[TABLE]

and it follows that $4\tau\bar{\tau}=3(c_{23}^{1})^{2}-4$.

Finally, by using (4.4) and (4.9), the second equation of (3.41) becomes

[TABLE]

This, if compared it with (3.22), gives

[TABLE]

On the other hand, using $\mu=0$ and (2.16), from (2.20) we obtain

[TABLE]

Then the last equation of (4.5) follows. ∎

Corollary 4.1**.**

Let $\varphi:S^{3}\rightarrow\mathbb{C}P^{2}$ be an equivariant CR minimal immersion.

(i)* If $\varphi(S^{3})$ is not a Berger sphere, then there is an orthonormal frame $\{\omega_{i}\}$ such that, in (4.6), it holds*

[TABLE]

(ii)* If $\varphi(S^{3})$ is a Berger sphere, then there is an orthonormal frame $\{\omega_{i}\}$ such that, in (4.6),*

[TABLE]

Proof.

(i) By (4.5) it holds $c_{12}^{1}=c_{13}^{1}=0$. Then, it follows from Proposition 3.1 that $\varphi(S^{3})$ is not a Berger sphere if and only if $\tau=c_{21}^{2}+\tfrac{i}{2}(c_{31}^{2}-c_{12}^{3})\not=0$. Then, by the third conclusion of Proposition 3.1, we can take a suitable frame transformation (3.23) such that $c_{23}^{1}<0$ and $\tau=|\tau|>0$. It follows from (4.5) that $c_{23}^{1}=-2,\ a=3$ and $\tau=\sqrt{2}$. Substituting these into (4.4) we obtain (4.11).

(ii) By replacing $X_{1}$ with $-X_{1}$ if necessary, we may assume that $c_{23}^{1}<0$. Then by (4.5) and $\tau=0$ we have $c_{23}^{1}=-2/\sqrt{3}$, $a=3\sqrt{3}$. Put these into (4.4) we get (4.12). ∎

The following proposition shows that $n=2$ is very exceptional, only in that case a non-Berger sphere could be admitted.

Proposition 4.1**.**

Any equivariant CR minimal immersion $\varphi:S^{3}\rightarrow\mathbb{C}P^{n}$ with $n\geq 3$ must be a Berger sphere.

Proof.

We consider two cases for $n\geq 3$:  (a) $\nabla^{\bot}\xi_{0}=0$;   (b) $\nabla^{\bot}\xi_{0}\neq 0$.

(a)  If $\nabla^{\bot}\xi_{0}=0$, then according to Proposition 3.2 and Lemma 4.1, we have $\lambda_{\alpha}=0$ and $\sum|\mu_{\alpha}|^{2}>0$ in (4.3). By re-choosing $e_{\alpha}$’s in the frame $\{e_{0},e_{1},e_{2},e_{\alpha}\}$ as stated in Lemma 3.1 we may assume that

[TABLE]

where $\mu_{3}\in\mathbb{R}$ is positive. By (3.38) we have $\sigma\wedge\omega=0$, and thus by (3.16) it holds $\mu=c_{12}^{1}+ic_{13}^{1}=0$ and $\tau=c_{21}^{2}+\tfrac{i}{2}(c_{31}^{2}-c_{12}^{3})=0$. It follows from Proposition 3.1 (ii) that $\varphi(S^{3})$ is a Berger sphere.

(b)  In this case, Proposition 3.2 implies that $\sum|\lambda_{\alpha}|^{2}>0$. Then, by re-choosing $e_{\alpha}$’s of $\{e_{0},e_{1},e_{2},e_{\alpha}\}$ we have constants $\lambda_{3}>0$ and $\mu_{4}\geq 0$ such that

[TABLE]

Using (3.35) and (3.38) we have

[TABLE]

where $i\rho_{3}=\theta_{33}$. Using (3.16) and comparing the coefficient of $\omega_{1}\wedge\bar{\omega}$ on both side of the above equation, we get $\tau=c_{21}^{2}+\tfrac{i}{2}(c_{31}^{2}-c_{12}^{3})=0$. Thus $c_{21}^{2}=-c_{31}^{3}=0$, $c_{31}^{2}=c_{12}^{3}$.

It follows by (3.16) and (4.2) that

[TABLE]

Substituting these into the second equation of (3.41) and then comparing the coefficient of $\omega_{1}\wedge\bar{\omega}$ with that in (3.22), we obtain

[TABLE]

On the other hand, from (2.20), and using (3.17) with the fact $\tau=0$, we get

[TABLE]

Then, combining with the previous equation, it implies that $c_{12}^{1}=c_{13}^{1}=0$.

Thus, as in case (a), $\varphi(S^{3})\subset\mathbb{C}P^{n}$ is a Berger sphere. ∎

The following lemma is also needed in Sect. 6.

Lemma 4.2**.**

Let $\varphi:S^{3}\rightarrow\mathbb{C}P^{n}$ be an equivariant CR minimal immersion with $n\geq 3$. Then, there exist positive real numbers $b$ and $c$, a normalized frame $\{\omega_{i}\}$ and a unitary frame $\{e_{0},e_{A}\}$ of $\underline{\mathbb{C}}^{n+1}$ such that $\varphi=[e_{0}]$ and

[TABLE]

[TABLE]

where $\lambda_{3}=\sqrt{1-3b^{2}c}\geq 0$. Moreover, $\nabla^{\perp}\xi_{0}=0$ if and only if $\lambda_{3}=0$.

Proof.

From the proof of Proposition 4.1, regardless of $\nabla^{\bot}\xi_{0}=0$ or $\nabla^{\bot}\xi_{0}\neq 0$, we can always choose a unitary frame $\{e_{0},e_{A}\}$ such that $\varphi=[e_{0}]$ and (4.14) holds.

Proposition 4.1 shows that $c_{12}^{1}=c_{13}^{1}=c_{21}^{2}=0$ and $c_{31}^{2}=c_{12}^{3}$, which implies that $\{\omega_{i}\}$ is a normalized frame. Next, following the proof of Corollary 4.1, we may assume $c_{23}^{1}<0$. Then, by Proposition 2.1, we see that $c:=c_{23}^{1}c_{31}^{2}/4$ and $b:=-c_{23}^{1}/(2\sqrt{c})$ are positive real numbers, and so that we have (4.15).

Now, (3.16) and the first equation of (3.19) become

[TABLE]

It follows that, by (3.33) and that $d$ is an isomorphism, $\rho_{0}=\tfrac{1}{b\sqrt{c}}\omega_{1}$. Hence, by (4.2), we have $\rho_{1}=\rho_{0}+c_{23}^{1}\omega_{1}=\tfrac{1-2b^{2}c}{b\sqrt{c}}\omega_{1}$.

From (4.14), (4.17) and the above, we can use the first equation of (3.36) to conclude that $\lambda_{3}^{2}=1-3b^{2}c$. Finally, the assertion that $\nabla^{\perp}\xi_{0}=0$ is equivalent to $\lambda_{3}=0$ follows from (4.16) and (3.32). ∎

If $n=2$, then by definition $\nabla^{\bot}\xi_{0}=0$ is trivially satisfied. Moreover, case (ii) in Corollary 4.1 can be looked as a special case of Lemma 4.2 with $c=4/3$, $b=\tfrac{1}{2}$ and $\lambda_{3}=0$. In summary, we have the following result.

Theorem 4.1**.**

Let $\varphi:S^{3}\rightarrow\mathbb{C}P^{n}$ be an equivariant CR minimal immersion with full dimension $n$. Then $\varphi$ has to be in one of the following three cases:

(1)* $n=2$ and $\varphi(S^{3})$ is not a Berger sphere case;*

(2)* $n\geq 2$ and $\varphi(S^{3})$ is a Berger sphere with $\nabla^{\bot}\xi_{0}=0$;*

(3)* $n\geq 3$ and $\varphi(S^{3})$ is a Berger sphere with $\nabla^{\bot}\xi_{0}\neq 0$.*

Remark 4.1*.*

In Theorem 4 of [11], Kim and Ryan proved that the only compact pseudo-Einstein hypersurfaces in $\mathbb{C}P^{2}$ are the geodesic spheres. We would point out that these geodesic spheres are Berger spheres of CR type (see, e.g., pp. 95-96 of [6]), and some of them are minimal depending on their radii.

5. Existence of equivariant CR minimal immersions

In this section we show that all the three cases in Theorem 4.1 do exist. Indeed, for each case the equivariant CR minimal immersions can be expressed explicitly in the form of polynomials. As the round sphere is a special Berger sphere, our results extend that of [12] we have mentioned in the introduction.

Let $\{X^{\prime}_{i}\}$ be the basis of $\mathfrak{su}(2)$ defined by (2.5), and set $Z^{\prime}=X^{\prime}_{2}+iX^{\prime}_{3}$. Then

[TABLE]

Let $\{\varepsilon_{0},\varepsilon_{1},\ldots,\varepsilon_{n}\}$ be the natural basis of $\mathbb{C}^{n+1}$ and $C_{\alpha}^{n}=n!/[\alpha!(n-\alpha)!]$ be the binomial coefficients. Define a $\mathbb{C}^{n+1}$-valued function

[TABLE]

and then set

[TABLE]

Applying Lemma 3.1 of [12] and the proof of Lemma 3.3 in [12], we have the following lemma, where $\{\omega^{\prime}_{i}\}$ is the dual frame of $\{X^{\prime}_{i}\}$ and $\omega^{\prime}=\omega^{\prime}_{2}+i\omega^{\prime}_{3}$.

Lemma 5.1**.**

For $n\geq 2$, $\{f_{0},f_{1},\ldots,f_{n}\}$ is a unitary frame of $\underline{\mathbb{C}}^{n+1}$. Moreover,

[TABLE]

for $\alpha=0,1,\ldots,n$.

5.1. Existence of equivariant CR minimal non-Berger

sphere

Define

[TABLE]

By Lemma 5.1, $\{f_{0},f_{1},f_{2}\}$ is a unitary frame of $\underline{\mathbb{C}}^{3}$. Moreover, it holds that

[TABLE]

Now, we define ${e_{0}}:{S^{3}}\rightarrow{\mathbb{C}^{3}}$ by

[TABLE]

Proposition 5.1**.**

The $\mathbb{C}^{3}$-valued function $e_{0}$, given by (5.4), defines an equivariant CR minimal immersion ${\varphi=[e_{0}]}:{S^{3}}\rightarrow{\mathbb{C}P^{2}}$, which is not a Berger sphere.

Proof.

Let $t=\pi/8$ and that

[TABLE]

Then, by virtue of Lemma 5.1, $\{e_{0},e_{1},e_{2}\}$ is a unitary frame of $\underline{\mathbb{C}}^{3}$. Solving (5.4) and (5.5), we get

[TABLE]

Using (5.3) – (5.5), and noting that $\sqrt{2}e^{2it}=1+i$, we have

[TABLE]

where $\omega_{1}=\sqrt{2}\omega^{\prime}_{1},\ \omega_{2}=\sqrt{2}\omega^{\prime}_{2}-\omega^{\prime}_{3},\ \omega_{3}=\omega^{\prime}_{3},\ \omega=\omega_{2}+i\omega_{3}$.

Then, by Proposition 4.2 of [12] we see that $\varphi$ is an equivariant immersion, with induced metric $ds^{2}=\sum\omega_{i}^{2}$ and an orthonormal frame $\{\omega_{i}\}$.

From (5.6) we know that $-\varphi^{*}\Omega=\tfrac{i}{2}\omega\wedge\bar{\omega}=\omega_{2}\wedge\omega_{3}$. Thus, $\varphi$ is of CR type.

Comparing (3.7) with (5.6), and noting that $\omega_{12}+i\omega_{13}=\sigma=-\theta_{12}$, we have

[TABLE]

From the above equations and (4.1) we see that $\varphi$ is minimal.

Finally, as $\omega\wedge\sigma=\sqrt{2}e^{2it}\omega\wedge\bar{\omega}\neq 0$, by (3.16), (3.17) and Proposition 3.1, we see that $(S^{3},ds^{2})$ is not a Berger sphere.

Moreover, from (5.6) and Proposition 3.2 (ii), it is easily seen that $\varphi(S^{3})$ is totally geodesic with respect to $W_{2}$. ∎

5.2. Existence of equivariant CR minimal Berger

sphere

In this subsection, we agree with the following range of indices

[TABLE]

whereas the indices $A,B,i,j$ having the convention as before.

Let $k,\ell$ be two integers with $k>\ell\geq 0$. Suppose $\{\varepsilon_{0},\ldots,\varepsilon_{k},\varepsilon^{\prime}_{0},\ldots,\varepsilon^{\prime}_{\ell}\}$ is the natural basis of $\mathbb{C}^{n+1}=\mathbb{C}^{k+1}\oplus\mathbb{C}^{\ell+1}$ with $n=k+\ell+1$ and $\varepsilon^{\prime}_{\alpha^{\prime}}=\varepsilon_{k+1+\alpha^{\prime}}$.

Define two functions $f:{S^{3}}\rightarrow{\mathbb{C}^{n+1}}$ and $h:{S^{3}}\rightarrow{\mathbb{C}^{n+1}}$ by

[TABLE]

According to Lemma 5.1, $\{f_{0},f_{1},\ldots,f_{k},h_{0},\ldots,h_{\ell}\}$ is a unitary frame of $\underline{\mathbb{C}}^{n+1}$.

For $t\in(0,\pi/2)$, we define a $\mathbb{C}^{n+1}$-valued function $e_{0}$ by

[TABLE]

Proposition 5.2**.**

For each $t\in(0,\pi/2)$, the function $e_{0}$, given by (5.7), defines an equivariant CR immersion $\varphi=[e_{0}]:S^{3}\rightarrow\mathbb{C}P^{n}$ with the following properties:

(i)* $(S^{3},ds^{2})$ is a Berger sphere;*

(ii)* $\nabla^{\bot}\xi_{0}=0$ if and only if $\ell=0$;*

(iii)* $\varphi$ is minimal if and only if $t$ satisfies that*

[TABLE]

Proof.

There is a unique $t_{1}\in[0,\pi/2)$ such that

[TABLE]

Now we put

[TABLE]

Then $\{e_{0},e_{A}\}$ is a unitary frame of $\underline{\mathbb{C}}^{n+1}$ due to that $\{f_{\alpha},h_{\alpha^{\prime}}\}$ is unitary.

From similar expressions as (5.2) we can see that $\langle{df_{\alpha}},{\bar{f}_{\beta}}\rangle$, $\langle{dh_{\alpha^{\prime}}},{\bar{h}_{\beta^{\prime}}}\rangle$ are all left-invariant one-forms, and that $\langle{df_{\alpha}},{\bar{h}_{\alpha^{\prime}}}\rangle=\langle{dh_{\alpha^{\prime}}},{\bar{f}_{\alpha}}\rangle=0$. It follows that $\langle{de_{0}},{\bar{e}_{0}}\rangle$, $\langle{de_{0}},{\bar{e}_{A}}\rangle$, $\langle{de_{A}},{\bar{e}_{B}}\rangle\in\mathfrak{su}(2)^{*}\otimes\mathbb{C}$. Hence $\varphi$ is equivariant.

From (5.7) and (5.10), with the use of (5.2) and (5.9), we can easily verify that

[TABLE]

where

[TABLE]

From (5.11), we see that $\varphi$ is an immersion with the induce metric

[TABLE]

Obviously, $ds^{2}$ is a Berger metric with $c=1/a^{\prime}_{0}$ and $b=m\sqrt{c}\,\sin 2t$ in (2.17).

Let $\{\omega_{i}\}$ be an orthonormal frame with respect to $ds^{2}$, where $\omega_{1}=m\sin 2t\kern 2.0pt\omega^{\prime}_{1}$, $\omega=\omega_{2}+i\omega_{3}=\sqrt{a^{\prime}_{0}}\kern 2.0pt\omega^{\prime}$. Then (5.11) can be rewritten as

[TABLE]

It follows that $-\varphi^{*}\Omega=(i/2)\omega\wedge\bar{\omega}=\omega_{2}\wedge\omega_{3}$. Thus $\varphi$ is of CR type.

Similarly, using (5.2), (5.9) and $de_{1}=i\sin t{\kern 2.0pt}df_{0}+\cos t{\kern 2.0pt}dh_{0}$, we can verify that

[TABLE]

where

[TABLE]

Then, Proposition 3.2 shows that $\nabla^{\bot}\xi_{0}=0$ if and only if $\ell=0$.

Finally, we have the calculation

[TABLE]

where $\mu_{4}=\sqrt{2k(k-1)}/a^{\prime}_{0}$, $\mu_{5}=\sqrt{2\ell(\ell-1)}/a^{\prime}_{0}$  (Note: $\mu_{5}=0$ if $\ell=0$).

From (5.15) and (5.17) we see that the second equation in (4.1) is satisfied. By using (5.14), (5.15) and (5.16), we get

[TABLE]

Therefore, by (4.1), $\varphi$ is minimal if and only if

[TABLE]

This leads to

[TABLE]

Then, noting that $k-\ell=2m$ and $k+\ell=n-1$, we can easily get (5.8).

We have completed the proof of Proposition 5.2. ∎

In the special case that $\ell=0$ and $k=n-1$, by (5.8) we have $t=\pi/6$ and (5.7) becomes

[TABLE]

From Proposition 5.2 and its proof we have the following

Corollary 5.1**.**

The function $e_{0}$, given by (5.20), defines an equivariant CR minimal immersion $\varphi=[e_{0}]:S^{3}\rightarrow\mathbb{C}P^{n}$, which induces a Berger metric $ds^{2}$. Moreover, it satisfies $\nabla^{\bot}\xi_{0}=0$.

Remark 5.1*.*

(a) The particular case $e_{0}=\tfrac{\sqrt{3}}{2}(z,w,0)+\tfrac{i}{2}(0,0,1)$, which corresponds to $n=2$ and $t=\pi/6$, is exactly the case (ii) of Corollary 4.1.

(b) From (5.13) one can see that the induced metric has constant curvature $c$ if and only if $\tfrac{1}{c}=m^{2}\sin^{2}2t=k\cos^{2}t+\ell\sin^{2}t=\tfrac{n-1}{2}+m\cos 2t$. This, by (5.18), is equivalent to $\cot 2t=\sqrt{c}$. If it does occur, then $m\cos 2t=1$, $n=2m^{2}-3$ and $c=1/(m^{2}-1)$ for some $m\geq 2$. We then recover Theorem 6.1 in [12].

(c) The function $e_{0}=e_{0}(t)$ in (5.7) depends on the parameter $t$. For a fixed pair $k,\ell$ with $k>\ell\geq 0$ there is a family of equivariant CR immersion $\{\varphi_{t}=[e_{0}(t)]\kern 1.0pt|\kern 1.0ptt\in(0,\pi/2)\}$. Each of them induces a Berger metric. However, among them only one is minimal by (5.8). Moreover, for fixed $n\geq 2$ there are $[n/2]$ such families, given by $\ell=0,\,\ldots,\,[n/2]-1$. Here, $[n/2]$ denotes the largest integer not exceeding $n/2$.

6. Uniqueness of equivariant CR minimal immersions

Recall that two immersions $\psi_{1},\psi_{2}:S^{3}\rightarrow\mathbb{C}P^{n}$ are said to be equivalent, denoted by $\psi_{1}\sim\psi_{2}$, if there exists a holomorphic isometry $A:\mathbb{C}P^{n}\rightarrow\mathbb{C}P^{n}$ such that $\psi_{1}=A\circ\psi_{2}$.

For simplicity, we denote by $\varphi_{1}$, $\varphi_{2}$, $\varphi_{3}$ the three equivariant CR minimal immersions determined by (5.4), (5.20), (5.7) with the additional conditions (5.8) and $\ell>0$, respectively. Let $X^{\prime}_{1},Z^{\prime}$ be the differential operators as defined by (5.1).

First of all, we introduce Lemma 3.3 in [12] as below.

Lemma 6.1** ([12]).**

Let $f$ be a $\mathbb{C}^{n+1}$-valued function defined on $S^{3}$. If it satisfies $\langle f,{\bar{f}}\rangle=1,\bar{Z}^{\prime}f=0,\,X^{\prime}_{1}f=isf$, then $s$ is a nonnegative integer with $s\leq n$. Moreover, there is a unitary basis $\{\varepsilon_{0},\ldots,\varepsilon_{s},\varepsilon_{s+1},\ldots,\varepsilon_{n}\}$ of $\mathbb{C}^{n+1}$ such that

[TABLE]

Next, we will prove two uniqueness theorems based on preceding Theorem 4.1. For that purpose, in sequel we assume that $\varphi:S^{3}\rightarrow\mathbb{C}P^{n}$ is an equivariant CR minimal immersion with full dimension $n$.

Theorem 6.1**.**

If $\varphi(S^{3})$ is not a Berger sphere, then up to an inner automorphism of $SU(2)$ we have $\varphi\sim\varphi_{1}=[e_{0}]$, where $e_{0}$ is defined by (5.4).

Proof.

According to Proposition LABEL:prop:4.2, we have $n=2$. Moreover, by Corollary 4.1 (i), we have a unitary frame $\{e_{0},e_{1},e_{2}\}$ of $\underline{\mathbb{C}}^{3}$ such that $\varphi=[e_{0}]$ and

[TABLE]

Now, we take another unitary frame $\{\tilde{e}_{0},\tilde{e}_{1},\tilde{e}_{2}\}$ of $\underline{\mathbb{C}}^{3}$ by

[TABLE]

or, equivalently,

[TABLE]

By setting $\omega^{\prime}_{1}=\frac{1}{\sqrt{2}}\omega_{1}$, $\omega^{\prime}=\omega^{\prime}_{2}+i\omega^{\prime}_{3}=\cos\frac{\pi}{8}\omega-i\sin\frac{\pi}{8}\bar{\omega}$, or equivalently,

[TABLE]

then, using that $\cos\frac{\pi}{8}+\sin\frac{\pi}{8}=\sqrt{2}\cos\frac{\pi}{8}$ and $\cos\frac{\pi}{8}-\sin\frac{\pi}{8}=\sqrt{2}\sin\frac{\pi}{8}$, we obtain $2\omega^{\prime}_{1}\wedge\omega^{\prime}_{2}\wedge\omega^{\prime}_{3}=\omega_{1}\wedge\omega_{2}\wedge\omega_{3}$. This shows that $\{\omega^{\prime}_{1},\omega^{\prime}_{2},\omega^{\prime}_{3}\}$ is a basis of $\mathfrak{su}(2)^{*}$.

Next, direct calculations show that

[TABLE]

On the other hand, exterior differentiations of (6.3) give that $d\omega^{\prime}_{1}=i\omega^{\prime}\wedge\bar{\omega}^{\prime}$, $d\omega^{\prime}=2i\omega^{\prime}_{1}\wedge\omega^{\prime}$. Hence $\{\omega^{\prime}_{i}\}$ satisfies (2.3). By Lemmas 2.1 and 2.2, up to an inner automorphism of $SU(2)$ we may assume that the dual frame $\{X^{\prime}_{i}\}$ of $\{\omega^{\prime}_{i}\}$ is defined by (2.5), or equivalently by (5.1). Then we have

[TABLE]

Using Lemma 6.1 we then have

[TABLE]

where $\{\varepsilon_{0},\varepsilon_{1},\varepsilon_{2}\}$ is a unitary basis of $\mathbb{C}^{3}$. By re-choosing a basis of $\mathbb{C}^{3}$, or equivalently by a holomorphic isometry $A:{\mathbb{C}^{3}}\rightarrow{\mathbb{C}^{3}}$, we may set

[TABLE]

Then, by (6.5), we have

[TABLE]

It follows from (6.3) that

[TABLE]

This completes the proof of Theorem 6.1. ∎

Theorem 6.2**.**

If $\varphi(S^{3})$ is a Berger sphere with $\nabla^{\bot}\xi_{0}=0$ (resp. with $\nabla^{\bot}\xi_{0}\neq 0$), then, up to an inner automorphism of $SU(2)$, we have $\varphi\sim\varphi_{2}=[e_{0}]$ and $e_{0}$ is defined by (5.20) ( resp. $\varphi\sim\varphi_{3}=[e_{0}]$ and $e_{0}$ is defined by (5.7) and (5.8) with some $\ell>0$).

Proof.

Since $\varphi(S^{3})$ is a Berger sphere, according to Corollary 4.1 (ii) and Lemma 4.2, regardless of $n=2$ or $n\geq 3$, we have a normalized frame $\{\omega_{i}\}$ and a unitary frame $\{e_{0},e_{A}\}$ such that $\varphi=[e_{0}]$ and

[TABLE]

[TABLE]

where $b,c$ are positive real numbers with $b=\tfrac{1}{2},c=\tfrac{4}{3}$ if $n=2$; $\lambda_{3}=\sqrt{1-3b^{2}c}\geq 0$ and that $\lambda_{3}=0$ if and only if $\nabla^{\bot}\xi_{0}=0$.

If we set

[TABLE]

then $\{\omega^{\prime}_{i}\}$ is a basis of $\mathfrak{su}(2)^{*}$. Using (6.6) we see that $\{\omega^{\prime}_{i}\}$ satisfy (2.3). As before, up to an inner automorphism of $SU(2)$, we may assume that the dual frame $\{X^{\prime}_{i}\}$ of $\{\omega^{\prime}_{i}\}$ is defined by (5.1).

From (6.7) and (6.8) we get

[TABLE]

where $\omega^{\prime}=\omega^{\prime}_{2}+i\omega^{\prime}_{3}$. Hence we have

[TABLE]

[TABLE]

Let $V$ be the subbundle of $\underline{\mathbb{C}}^{n+1}$ spanned by $\{e_{0},e_{1}\}$. Then (6.11) defines an endomorphism ${X_{1}}:V\rightarrow V$. It is skew-Hermitian, and therefore has two imaginary eigenvalues $ik,i\ell$ such that

[TABLE]

Since $k>c^{-1}>0$ and $k\ell=c^{-2}(1-3b^{2}c)\geq 0$, we see that $k>\ell\geq 0$, and $\ell=0$ if and only if $\lambda_{3}=0$, or equivalently, $\nabla^{\bot}\xi_{0}=0$.

It is well known that there is a unitary frame $\{f,h\}$ of $V$ such that

[TABLE]

Since by theory of linear algebra $(X^{\prime}_{1}-ik\kern 2.0ptI)(X^{\prime}_{1}-i\ell\kern 2.0ptI)=0$, and, by (6.11) and (6.12), we have

[TABLE]

[TABLE]

We may choose $\{f,h\}$ such that

[TABLE]

where $t\in(0,\pi/2)$ is determined by

[TABLE]

Then, $|f|=|h|=1$, $\langle f,{\bar{h}}\rangle=0$. Moreover, by (6.10) and (6.13), we get

[TABLE]

Since $n$ is the full dimension, applying Lemma 6.1 we see that $k,\ell$ are integers with $k>\ell\geq 0$ and $k+\ell=n-1$. Moreover, there is a unitary basis $\{\varepsilon_{0},\ldots,\varepsilon_{k},\varepsilon^{\prime}_{0},\ldots,\varepsilon^{\prime}_{\ell}\}$ of $\mathbb{C}^{n+1}$ such that

[TABLE]

Thus, by (6.14), we can solve $e_{0}$ to obtain that

[TABLE]

If $\nabla^{\bot}\xi_{0}=0$, then $\lambda_{3}=0$ and thus $b=1/\sqrt{3c}$. Hence $\ell=0$, $t=\pi/6$. Then $\varphi\sim\varphi_{2}$, due to that (6.16) differs from (5.20) by a holomorphic isometry of $\mathbb{C}^{n+1}$.

If $\nabla^{\bot}\xi_{0}\neq 0$, then, by (6.12), $k>\ell>0$. Since $\varphi$ is minimal, according to $(c)$ of Remark 5.1, the parameter $t$ determined by (6.15) has to be that one determined by (5.8). Then, using (6.12) and (6.15), one gets

[TABLE]

Now, by comparing (6.16) with (5.7), we come to the conclusion $\varphi\sim\varphi_{3}$. ∎

Completion of proof of Theorem 1.1.

Let $\varphi:{S^{3}}\rightarrow{\mathbb{C}}P^{n}\ (n\geq 2)$ be a linearly full equivariant CR minimal immersion with induced metric $ds^{2}$. If $(S^{3},ds^{2})$ is not a Berger sphere, then according to Theorem 6.1, we get the assertion (1). If on the other hand $(S^{3},ds^{2})$ is a Berger sphere, then Theorem 6.2 shows that for both cases of $\nabla^{\bot}\xi_{0}=0$ and $\nabla^{\bot}\xi_{0}\neq 0$, the expression of $\varphi$ assume the form as stated in the assertion (2), where $\nabla^{\bot}\xi_{0}=0$ corresponds to $\ell=0$, and $\nabla^{\bot}\xi_{0}\neq 0$ corresponds to an integer $\ell>0$. ∎