Cyclic parallel CR-submanifolds of maximal CR-dimension in a complex space form

Abstract

We first classify $(2n-1)$-dimensional cyclic parallel CR-submanifold $M$ with CR-dimension $n-1$ in a non-flat complex space form of constant holomorphic sectional curvature $4c$. Then, we prove that $||\nabla h||^2\ge 4(n-1)c^2$, where $h$ is the second fundamental form on $M$. We also completely classify $(2n-1)$-dimensional CR-submanifolds with CR-dimension $n-1$ in a non-flat complex space form which satisfy the equality case of this inequality. This generalizes an inequality for real hypersurfaces in a non-flat complex space form obtained by Maeda (J Math Soc Jpn 28:529–540; 1976) and Chen et al. (Algebras Groups Geom 1:176–212; 1984) for complex projective and hyperbolic spaces, respectively.

1 Introduction

A complex $n$-dimensional complex space form $\hat{M}_n(c)$ is a complete and simply connected Kaehler manifold with constant holomorphic sectional curvature $4c$, that is, it is either a complex projective space $\mathbb{C }P_n$, a complex Euclidean space $\mathbb{C }_n$, or a complex hyperbolic space $\mathbb{C }H_n$ (according to as the holomorphic sectional curvature $4c$ is positive, zero, or negative).

The study of real hypersurfaces in a Kaehler manifold has been an active field in the past few decades, especially when the ambient space is a complex space form. One of the first results in this topic is the non-existence of real hypersurfaces $M$ with parallel shape operator $A$ in a non-flat complex space form, that is, $\nabla A=0$, where $\nabla$ is the Levi-Civita connection on $M$. This fact is an immediate consequence of the Codazzi equation of such a submanifold. Several weaker notions such as $\eta$-parallelism and recurrence of the shape operator were hence studied by the researchers.

The shape operator $A$ is said to be recurrent if there is a $1$-form $\tau$ on $M$ such that $\nabla A=A\otimes \tau$. It is known that there does not exist any real hypersurface in $\hat{M}_n(c),\,c\ne 0$, with recurrent shape operator (cf. [14, 21]). A real hypersurface $M$ in $\hat{M}_n(c)$ is said to be $\eta$-recurrent if $\langle \nabla _XA)Y,Z\rangle =\tau (X)\langle AY,Z\rangle$, for any tangent vector fields $X, Y, and Z$ in the maximal holomorphic distribution ${\fancyscript{D}}$, where $\tau$ is a $1$-form on $M$ (cf. [13]). In particular, $M$ is said to be $\eta$-parallel when $\tau =0$ (cf. [17]).

In [18, 19], the author and Kon classified $\eta$-parallel real hypersurfaces in $\hat{M}_n(c),\,c\ne 0,\,n\ge 3$. It was also proved in [20] that a real hypersurface in $\hat{M}_n(c),\,c\ne 0,\,n\ge 3$ is $\eta$-recurrent if and only if it is $\eta$-parallel.

A submanifold $M$ in a Riemannian manifold $\hat{M}$ is said to be cyclic parallel if its second fundamental form $h$ satisfies

$$\begin{aligned} (\nabla _Xh)(Y,Z)+(\nabla _Y)h(Z,X)+(\nabla _Zh)(X,Y)=0 \end{aligned}$$

for any vector fields $X, Y$, and $Z$ tangent to $M$. When $M$ is a real hypersurface in $\hat{M}_n(c)$, the cyclic parallelism is equivalent to the condition

$$\begin{aligned} (\nabla _X A)Y=-c\{\eta (Y)\phi X+\langle \phi X,Y\rangle \xi \} \end{aligned}$$

for any vector fields $X$ and $Y$ tangent to $M$, where $(\phi ,\xi ,\eta ,\langle ,\rangle )$ is the almost contact structure on $M$ induced by the complex structure $J$ of the ambient space. Maeda (cf. [22]) and Chen, Ludden and Montiel (cf. [5]) classified real hypersurfaces in $\hat{M}_n(c),\,c\ne 0$, under this condition (cf. Theorem 4). With this result, it can be proved that

$$\begin{aligned} ||\nabla A||^2\ge 4(n-1)c^2 \end{aligned}$$

(1)

and equality holds if and only if the real hypersurface $M$ is an open part of a tube over $\mathbb{C }P_k,\,1\le k\le n-1$, for $c>0$, and $M$ is an open part of a horosphere, a geodesic hypersphere in $\mathbb{C }H_n$, or a tube over $\mathbb{C }H_k,\,1\le k\le n-1$, for $c<0$.

Note that a real hypersurface in $\hat{M}_n(c)$ is a CR-submanifold (see Definition 2 for precise definition) of maximal CR-dimension (or of hypersurface type). Hence, one of the main lines deals with generalizing these known results in real hypersurfaces in $M_n(c)$ to CR-submanifolds of maximal CR-dimension in $\hat{M}_n(c)$. A number of results were obtained by Djorić and Okumura (cf. [7]–[11]). In particular, they attempted to generalize certain results concerning relationship between $A$ and $\phi$ for real hypersurfaces in a complex space form into the setting of CR-submanifolds of maximal CR-dimension.

This paper is also a contribution in this line. The main objective of this paper is to extend the inequality (1) for real hypersurfaces in a non-flat complex space form to the setting of CR-submanifolds of maximal CR-dimension. We shall first prove the following theorem.

Theorem 1

Let $M$ be a $(2n-1)$-dimensional CR-submanifold of maximal CR-dimension in $\hat{M}_{n+p}(c),\,c\ne 0,\,n\ge 2$. Then, $M$ is cyclic parallel if and only if $M$ is an open part of one of the following spaces.

• (a) For $c<0$:

  • (i) a horosphere in $\mathbb{C }H_n$,

  • (ii) a geodesic hypersphere or a tube over a hyperplane $\mathbb{C }H_{n-1}$ in $\mathbb{C }H_n$,

  • (iii) a tube over a totally geodesic $\mathbb{C }H_{k}$ in $\mathbb{C }H_n$, where $1\le k\le n-2$.

• (b) For $c>0$:

  • (i) a geodesic hypersphere in $\mathbb{C }P_n$,

  • (ii) a tube over a totally geodesic $\mathbb{C }P_{k}$ in $\mathbb{C }P_n$, where $1\le k\le n-2$,

  • (iii) a standard CR-product $\mathbb{C }P_{n-1}\times \mathbb{R }P^1$ in $\mathbb{C }P_{2n-1}$.

With this result, we can prove the following.

Theorem 2

Let $M$ be a $(2n-1)$-dimensional CR-submanifold of maximal CR-dimension in $\hat{M}_{n+p}(c),\,c\ne 0,\,n\ge 2$. Then, $M$ satisfies

$$\begin{aligned} ||\nabla h||^2\ge 4(n-1)c^2 \end{aligned}$$

and equality holds if and only $M$ is an open part of one of the following spaces.

• (a) For $c<0$:

  • (i) a horosphere in $\mathbb{C }H_n$,

  • (ii) a geodesic hypersphere or a tube over a hyperplane $\mathbb{C }H_{n-1}$ in $\mathbb{C }H_n$,

  • (iii) a tube over a totally geodesic $\mathbb{C }H_{k}$ in $\mathbb{C }H_n$, where $1\le k\le n-2$.

• (b) For $c>0$:

  • (i) a geodesic hypersphere in $\mathbb{C }P_n$,

  • (ii) a tube over a totally geodesic $\mathbb{C }P_{k}$ in $\mathbb{C }P_n$, where $1\le k\le n-2$,

  • (iii) a standard CR-product $\mathbb{C }P_{n-1}\times \mathbb{R }P^1$ in $\mathbb{C }P_{2n-1}$.

Remark 1

It is worthwhile to remark that there is an additional class of submanifolds, that is, $\mathbb{C }P_{n-1}\times \mathbb{R }P^1$ in Case (b)(iii), appeared in the list of Theorem 1 compared to the classification of real hypersurfaces under the same condition (cf. Theorem 4). Chen and Maeda (cf. [6]) proved that there do not exist real hypersurfaces which are Riemannian product of Riemannian manifolds. Hence, we can see that $\mathbb{C }P_{n-1}\times \mathbb{R }P^1$ can never be immersed in $\mathbb{C }P_n$ as a real hypersurface.

This paper is organized as follows. In the next two sections, we shall fix some notations and discuss some fundamental properties of CR-submanifolds in a Kaehler manifold. We describe the standard examples of cyclic parallel CR-submanifolds of maximal CR-dimension in a non-flat complex space form in Sect. 4. In Sect. 5, we prepare some lemmas. We prove Theorem 1 and Theorem 2 in the last two sections.

2 CR-submanifolds in a Kaehler manifold

In this section, we shall recall some structural equations in the theory of CR-submanifolds in a Kaehler manifold and fix some notations. Some fundamental properties of CR-submanifolds in a Kaehler manifold are also derived here.

Let $\hat{M}$ be a Kaehler manifold with complex structure $J$, and let $M$ be a connected Riemannian manifold isometrically immersed in $\hat{M}$. The maximal $J$-invariant subspace ${\fancyscript{D}}_x$ of the tangent space $T_xM,\,x\in M$ is given by

$$\begin{aligned} {\fancyscript{D}}_x=T_xM\cap JT_xM. \end{aligned}$$

Definition 1

([4]) A submanifold $M$ in a Kaehler manifold $\hat{M}$ is said to be a generic submanifold if the dimension of ${\fancyscript{D}}_x$ is constant along $M$. The distribution ${\fancyscript{D}} : x\rightarrow {\fancyscript{D}}_x,\,x\in M$ is called the holomorphic distribution (or Levi distribution) on $M$ and the complex dimension of ${\fancyscript{D}}$ is called the CR-dimension of $M$.

Definition 2

([1]) A generic submanifold $M$ in a Kaehler manifold $\hat{M}$ is said to be a CR-submanifold if the orthogonal complementary distribution ${\fancyscript{D}}^\perp$ of ${\fancyscript{D}}$ in $TM$ is totally real, that is, $J{\fancyscript{D}}^\perp _x\subset T_xM^\perp ,\,x\in M$.

If ${\fancyscript{D}}^\perp =\{0\}$ (resp. ${\fancyscript{D}}=\{0\}$), the CR-submanifold $M$ is said to be holomorphic (resp. totally real). A CR-submanifold $M$ is said to be proper if it is neither holomorphic nor totally real. Let $\nu$ be the orthogonal complementary distribution of $J{\fancyscript{D}}^\perp$ in $TM^\perp$. Then, an anti-holomorphic submanifold $M$ is a CR-submanifold with $\nu =\{0\}$, that is, $J{\fancyscript{D}}^\perp =TM^\perp$.

Remark 2

The study of CR-submanifolds in the sense of Definition 2 was initiated by Bejancu in [1]. Generic submanifolds have been studied by some researchers under the term of “CR-submanifolds” from the CR geometric view point (cf. [12, 24, pp. 345]). We will not follow this term here in order to avoid the confusion. We remark that when a generic submanifold $M$ is of maximal CR-dimension, that is, $\dim _{\mathbb{R }}{\fancyscript{D}}=\dim M-1,\,M$ will be a CR-submanifold in the sense of Definition 2.

Suppose $M$ is a CR-submanifold in a Kaehler manifold $\hat{M}$. Denote by $\langle ,\rangle$ the Riemannian metric of $\hat{M}$ as well as that induced on $M$. Also, we let $\nabla$ be the Levi-Civita connection on the tangent bundle $TM$ of $M,\,\nabla ^\perp$ the normal connection on the normal bundle $TM^\perp$ of $M,\,h$ the second fundamental form, and $A_\sigma$ the shape operator of $M$ with respect to a vector $\sigma$ normal to $M$.

For a vector bundle $\fancyscript{V}$ over $M$, we denote by $\varGamma (\fancyscript{V})$ the $\Omega ^0(M)$-module of cross sections on $\fancyscript{V}$, where $\Omega ^k(M)$ is the space of $k$-forms on $M$. For any $X\in \varGamma (TM)$ and $\sigma \in \varGamma (TM^\perp )$, we put $\phi X=\tan (JX),\,\omega X=\text{ nor}(JX),\,B\sigma =\tan (J\sigma )$ and $C\sigma =\text{ nor}(J\sigma )$. From the parallelism of $J$, we have (cf. [27, pp. 77])

$$\begin{aligned} (\nabla _X\phi )Y&= A_{\omega Y}X+Bh(X,Y)\end{aligned}$$

(2)

$$\begin{aligned} (\nabla _X\omega )Y&= -h(X,\phi Y) +Ch(X,Y) \end{aligned}$$

(3)

$$\begin{aligned} (\nabla _XB)\sigma&= -\phi A_{\sigma }X+A_{C\sigma }X \end{aligned}$$

(4)

$$\begin{aligned} (\nabla _XC)\sigma&= -\omega A_{\sigma }X-h(X,B\sigma ) \end{aligned}$$

(5)

for any $X,\,Y\in \varGamma (TM)$ and $\sigma \in \varGamma (TM^\perp )$.

We denote by $H:=\text{ Trace}(h)$. For a local frame of orthonormal vectors $e_1,e_2,\ldots ,e_{2m}$ in $\varGamma ({\fancyscript{D}})$, where $m=\dim _{\mathbb{C }}{\fancyscript{D}}$, we define

$$\begin{aligned} H_{{\fancyscript{D}}}:=\sum ^{2m}_{j=1}h(e_j,e_j). \end{aligned}$$

Lemma 1

Let $M$ be a CR-submanifold in a Kaehler manifold $\hat{M}$. Then, $\langle (\phi A_\sigma +A_\sigma \phi )X,Y\rangle =0$, for any $X,Y\in \varGamma ({\fancyscript{D}})$ and $\sigma \in \varGamma (\nu )$. Moreover, we have $CH_{{\fancyscript{D}}}=0$.

Proof

By putting $X, Y\in \varGamma ({\fancyscript{D}})$ in (3), we have

$$\begin{aligned} -\omega \nabla _XY=-h(X,\phi Y)+Ch(X,Y). \end{aligned}$$

Taking inner product of both sides of this equation with $\sigma \in \varGamma (\nu )$, we obtain

$$\begin{aligned} 0=\langle \phi A_\sigma X,Y\rangle -\langle A_{C\sigma }X,Y\rangle . \end{aligned}$$

Since $A_{C\sigma }$ is self-adjoint, we obtain $\langle (\phi A_\sigma +A_\sigma \phi )X,Y\rangle =0$, for any $X,Y\in \varGamma ({\fancyscript{D}})$. Furthermore, for any unit vector field $X\in \varGamma ({\fancyscript{D}})$ and $\sigma \in \varGamma (\nu )$, we have

$$\begin{aligned} 0=\langle (\phi A_\sigma +A_\sigma \phi )X,\phi X\rangle =\langle h(X,X)+h(\phi X,\phi X),\sigma \rangle . \end{aligned}$$

This equation implies that $\langle H_{{\fancyscript{D}}},\sigma \rangle =0$ and hence $CH_{{\fancyscript{D}}}=0$. $\square$

A CR-submanifold $M$ is said to be mixed totally geodesic if $h(X,Y)=0$, for any $X\in \varGamma ({\fancyscript{D}})$ and $Y\in \varGamma ({\fancyscript{D}}^\perp )$. A CR-submanifold $M$ is called a CR-product if it is locally a Riemannian product of a holomorphic submanifold and a totally real submanifold.

The following lemma characterizes CR-products in a Kaehler manifold.

Lemma 2

([3]) A CR-submanifold $M$ in a Kaehler manifold is a CR-product if and only if $Bh(X,Y)=0$, for any $X\in \varGamma ({\fancyscript{D}})$ and $Y\in \varGamma (TM)$.

Now suppose $\hat{M}_q(c)$ is a $q$-dimensional complex space form with constant holomorphic sectional curvature $4c$, and let $M$ be a CR-submanifold in $\hat{M}_q(c)$.

Let $R$ and $R^\perp$ be the curvature tensors associated with $\nabla$ and $\nabla ^\perp$, respectively. The equations of Gauss, Codazzi, and Ricci are then given, respectively, by

$$\begin{aligned} R(X,Y)Z&= c \{\langle Y,Z\rangle X- \langle X,Z\rangle Y+ \langle \phi Y,Z\rangle \phi X-\langle \phi X,Z\rangle \phi Y \nonumber \\&\quad -2\langle \phi X,Y\rangle \phi Z \} +A_{h(Y,Z)}X-A_{h(X,Z)}Y\\&(\nabla _{X}h)(Y,Z)-(\nabla _{Y}h)(X,Z) =c\{\langle \phi Y,Z\rangle \omega X-\langle \phi X,Z\rangle \omega Y - 2\langle \phi X,Y\rangle \omega Z\}\nonumber \\&R^\perp (X,Y)\sigma =c\{\langle \omega Y,\sigma \rangle \omega X-\langle \omega X,\sigma \rangle \omega Y-2\langle \phi X,Y\rangle C\sigma \} +h(X,A_\sigma Y)-h(Y,A_\sigma X)\nonumber \end{aligned}$$

(6)

for any $X,\,Y,\,Z\in \varGamma (TM)$ and $\sigma \in \varGamma (TM^\perp )$.

A submanifold $M$ in a Riemannian manifold $\hat{M}$ is said to be cyclic parallel if its second fundamental form $h$ satisfies

$$\begin{aligned} (\nabla _Xh)(Y,Z)+(\nabla _Y)h(Z,X)+(\nabla _Zh)(X,Y)=0 \end{aligned}$$

for any $X, Y$, and $Z\in \varGamma (TM)$. When $M$ is CR-submanifold in $\hat{M}_q(c)$, by the Codazzi equation, the cyclic parallelism of $M$ is equivalent to the condition

$$\begin{aligned} (\nabla _Xh)(Y,Z)=-c\{\langle \phi X,Z\rangle \omega Y+\langle \phi X,Y\rangle \omega Z\} \end{aligned}$$

(7)

for any $X, Y$, and $Z\in \varGamma (TM)$.

The second-order covariant derivative $\nabla ^2 h$ on the second fundamental form $h$ is defined by

$$\begin{aligned} (\nabla ^2_{XY} h)(Z,W)&= \nabla _X^\perp \{(\nabla _Yh)(Z,W)\}-(\nabla _{\nabla _XY}h)(Z,W)-(\nabla _Yh)(\nabla _XZ,W)\\&-(\nabla _Yh)(Z,\nabla _XW). \end{aligned}$$

The Ricci identity gives

$$\begin{aligned} R(X,Y)h=\nabla ^2_{XY} h-\nabla ^2_{YX} h \end{aligned}$$

(8)

where

$$\begin{aligned} (R(X,Y)h)(Z,W)=R^\perp (X,Y)h(Z,W)-h(R(X,Y)Z,W)-h(Z,R(X,Y)W) \end{aligned}$$

for any $X, Y, Z$, and $W\in \varGamma (TM)$.

Finally, we state without proof a codimension reduction theorem for real submanifolds in a non-flat complex space form.

Theorem 3

([15, 25]) Let $M$ be a connected real $n$-dimensional submanifold in $\hat{M}_{(n+p)/2}(c),\, c\ne 0$ and let $N_0(x)$ be the orthogonal complement of the first normal space in $T_xM^\perp$. We put $H_0(x)=JN_0(x)\cap N_0(x)$ and let $H(x)$ be a $J$-invariant subspace of $H_0(x)$. If the orthogonal complement $H_2(x)$ of $H(x)$ in $T_xM^\perp$ is invariant under parallel translation with respect to the normal connection and if $q$ is the constant dimension of $H_2(x)$, for each $x\in M$, then there exists a $(n+q)$-dimensional totally geodesic holomorphic submanifold $\hat{M}_{(n+q)/2}(c)$ in $\hat{M}_{(n+p)/2}(c)$ such that $M\subset \hat{M}_{(n+q)/2}(c)$.

3 CR-submanifolds of maximal CR-dimension in a complex space form

Suppose $\hat{M}_{n+p}(c)$ is a complex $(n+p)$-dimensional complex space form of constant holomorphic sectional curvature $4c$, and $M$ is a real $(2n-1)$-dimensional CR-submanifold of maximal CR-dimension in $\hat{M}_{n+p}(c)$. Then, $\dim _{\mathbb{C }}{\fancyscript{D}}=n-1$ and $\dim {\fancyscript{D}}^\perp =1$. Let $N\in \varGamma (J{\fancyscript{D}}^\perp )$ be a local unit vector field normal to $M,\,\xi =-JN$ and $\eta$ the 1-form dual to $\xi$. Then, we have

$$\begin{aligned}&\phi ^2X =-X+\eta (X)\xi \\&\omega X =\eta (X)N; \quad B\sigma =-\langle \sigma ,N\rangle \xi \end{aligned}$$

for any $X\in \varGamma (TM)$ and $\sigma \in \varGamma (TM^\perp )$. It follows from (2)–(5) that

$$\begin{aligned} (\nabla _X\phi )Y&= \eta (Y)A_NX-\langle A_NX,Y\rangle \xi \end{aligned}$$

(9)

$$\begin{aligned} \nabla _X\xi&= \phi A_NX; \quad \nabla ^\perp _XN=Ch(X,\xi ) \end{aligned}$$

(10)

$$\begin{aligned} h(X,\phi Y)&= -\langle \phi A_NX,Y\rangle N-\eta (Y)Ch(X,\xi )+Ch(X,Y) \end{aligned}$$

(11)

$$\begin{aligned} (\nabla _XC)\sigma&= -\langle h(X,\xi ),\sigma \rangle N+\langle \sigma ,N\rangle h(X,\xi ) \end{aligned}$$

(12)

for any $X,\,Y\in \varGamma (TM)$ and $\sigma \in \varGamma (TM^\perp )$.

The equations of Codazzi and Ricci can also be reduced to

$$\begin{aligned} (\nabla _{X}h)(Y,Z)\!-\!(\nabla _{Y}h)(X,Z)&\!=\!c\{\eta (X)\langle \phi Y,Z\rangle \!-\!\eta (Y)\langle \phi X,Z\rangle \!-\! 2\eta (Z)\langle \phi X,Y\rangle \}N \end{aligned}$$

(13)

$$\begin{aligned} R^\perp (X,Y)\sigma&\!=\!-\!2c\langle \phi X,Y\rangle C\sigma \!+\!h(X,A_\sigma Y)\!-\!h(Y,A_\sigma X) \end{aligned}$$

(14)

for any $X,\,Y,\,Z\in \varGamma (TM)$ and $\sigma \in \varGamma (TM^\perp )$. We define the covariant derivative of the shape operator as

$$\begin{aligned} (\nabla _XA)_\sigma Y=\nabla _X\{A_\sigma Y\}-A_{\sigma }\nabla _XY-A_{\nabla ^\perp _X\sigma }Y. \end{aligned}$$

(15)

Then, we have

$$\begin{aligned} \langle (\nabla _XA)_\sigma Y,Z\rangle =\langle (\nabla _Xh)(Y,Z),\sigma \rangle \end{aligned}$$

and the Codazzi equation (13) can be rephrased as

$$\begin{aligned} (\nabla _{X}A)_\sigma Y-(\nabla _{Y}A)_\sigma X&= c\langle \sigma ,N\rangle \{\eta (X)\phi Y-\eta (Y)\phi X- 2\langle \phi X,Y\rangle \xi \} \end{aligned}$$

(16)

for any $X,\,Y,\,Z\in \varGamma (TM)$ and $\sigma \in \varGamma (TM^\perp )$.

The following lemma can be obtained immediately from Lemma 1.

Lemma 3

Let $M$ be a CR-submanifold of maximal CR-dimension in a Kaehler manifold $\hat{M}$. Then, $Ch(\xi ,\xi )=CH$.

4 Examples

In this section, we discuss certain examples of cyclic parallel CR-submanifolds of maximal CR-dimension in a non-flat complex space form. From (7), it is equivalent to said that $M$ satisfies the following condition.

$$\begin{aligned} (\nabla _Xh)(Y,Z)=-c\{\eta (Y)\langle \phi X,Z\rangle +\eta (Z)\langle \phi X,Y\rangle \}N \end{aligned}$$

(17)

for any $X, Y$, and $Z\in \varGamma (TM)$.

Let $\mathbb{C }_{n+1}^1$ be the complex Lorentzian space with Hermitian inner product

$$\begin{aligned} G(z,w)=-z_0\bar{w}_0+\sum ^n_{j=1}z_j\bar{w}_j \end{aligned}$$

where $z=(z_0,z_1,\ldots ,z_n),w=(w_0,w_1,\ldots ,w_n)\in \mathbb{C }_{n+1}^1$. Then, the anti-De Sitter space of radius $1$ is given by

$$\begin{aligned} H^{2n+1}_1:=H^{2n+1}_1(-1)=\left\{ z\in {\mathbb{C }}_{n+1}^1 : \langle z,z\rangle =-1\right\} \end{aligned}$$

where $\langle z,w\rangle :=\mathfrak R G(z,w)$. We denote by $\psi :H^{2n+1}_1\rightarrow \mathbb{C }H_n$ the principal $S^1$-bundle over $\mathbb{C }H_n$. Here, $\mathbb{C }H_n$ denotes the complex hyperbolic space with constant holomorphic sectional curvature $-4$.

Example 1

(Horospheres in $\mathbb{C }H_n$) Let $M^{\prime }$ be a Lorentzian hypersurface in $H^{2n+1}_1$ given by

$$\begin{aligned} |z_0-z_1|=1; \qquad -|z_0|+\sum ^{n}_{j=1}|z_j|^2=-1. \end{aligned}$$

Then, $M^*=\psi (M^{\prime })$ is a real hypersurface in $\mathbb{C }H_n$, so-called a horosphere (a self-tube).

Example 2

(Tubes over $\mathbb{C }H_k$ in $\mathbb{C }H_n,\,0\le k\le n-1$) Let $k,l\ge 0$ be integers with $k+l=n-1,\,r>0$. We consider a Lorentzian hypersurface $M^{\prime }_k(r)$ in $H^{2n+1}_1$ defined by

$$\begin{aligned} -|z_0|^2+\sum ^k_{j=1}|z_j|^2=-{\cosh }^2 r, \qquad -|z_0|+\sum ^{n}_{j=1}|z_j|^2=-1. \end{aligned}$$

Then, $M^{\prime }_{k}(r)$ is the standard product $H^{2k+1}_1(-\cosh r)\times S^{2l+1}(\sinh r)$. $M_k(r)=\psi (M^{\prime }_k(r))$ is a real hypersurface in $\mathbb{C }H_n$, which is a tube of radius $r$ over a totally geodesic holomorphic submanifold $\mathbb{C }H_k$ in $\mathbb{C }H_n$. In particular, $M_k(r)$ is a geodesic hypersphere in $\mathbb{C }H_n$ when $k=0$.

Now, we consider the complex Euclidean space $\mathbb{C }_{n+1}$ with Hermitian inner product

$$\begin{aligned} G(z,w)=\sum ^n_{j=0}z_j\bar{w}_j \end{aligned}$$

where $z=(z_0,z_1,\ldots ,z_n),w=(w_0,w_1,\ldots ,w_n)\in \mathbb{C }_{n+1}$. Then, the sphere of radius $1$ centered at the origin is given by

$$\begin{aligned} S^{2n+1}:=S^{2n+1}(1)=\{z\in \mathbb{C }_{n+1} : \langle z,z\rangle =1\} \end{aligned}$$

where $\langle z,w\rangle :=\mathfrak R G(z,w)$. We denote by $\psi :S^{2n+1}\rightarrow \mathbb{C }P_n$ the principal $S^1$-bundle over $\mathbb{C }P_n$. Here, $\mathbb{C }P_n$ denotes the complex projective space with constant holomorphic sectional curvature $4$.

Example 3

(Tubes over $\mathbb{C }P_k$ in $\mathbb{C }P_n,\,0\le k\le n-1$) Let $k,l\ge 0$ be integers with $k+l =n-1,\,r\in \,]\,0,\pi /2[$. We consider a hypersurface $M^{\prime }_k(r)$ in $S^{2n+1}$ defined by

$$\begin{aligned} \sum ^k_{j=0}|z_j|^2={\cos }^2 r, \qquad \sum ^{n}_{j=0}|z_j|^2=1. \end{aligned}$$

Then, $M^{\prime }_{k}(r)$ is the standard product $S^{2k+1}(\cos r)\times S^{2l+1}(\sin r)$. $M_k(r)=\psi (M^{\prime }_k(r))$ is a real hypersurface in $\mathbb{C }P_n$, which is a tube of radius $r$ over a totally geodesic holomorphic submanifold $\mathbb{C }P_k$ in $\mathbb{C }P_n$. In particular, when $k=0,\,M_k(r)$ is a geodesic hypersphere in $\mathbb{C }P_n$.

Theorem 4

[5, 22] Let $M$ be a real hypersurface in $\hat{M}_n(c),\,c\ne 0,\,n\ge 2$. Then, $M$ satisfies

$$\begin{aligned} (\nabla _XA)Y=-c\{\eta (Y)\phi X+\langle \phi X,Y\rangle \xi \} \end{aligned}$$

for any $X, Y\in \varGamma (TM)$, if and only if $M$ is an open part of one of the following spaces.

• (a) For $c<0$

  • (i) a horosphere,

  • (ii) a geodesic hypersphere or a tube over $\mathbb{C }H_{n-1}$,

  • (iii) a tube over $\mathbb{C }H_{k}$, where $1\le k\le n-2$.

• (b) For $c>0$

  • (i) a geodesic hypersphere,

  • (ii) a tube over $\mathbb{C }P_{k}$, where $1\le k\le n-2$.

Remark 3

A real hypersurface in a Kaehler manifold is said to be Hopf if it is mixed totally geodesic. The real hypersurfaces stated in Theorem 4 are categorized as Hopf hypersurfaces of type $A$ in the Takagi’s list (for $c>0$) and Montiel’s list (for $c<0$) of Hopf hypersurfaces of constant principal curvatures in $\hat{M}_n(c),\,c\ne 0$ (cf. [23, 26]). These real hypersurfaces in the Takagi’s list and Montiel’s list are in fact the only Hopf hypersurfaces with constant principal curvatures in $\hat{M}_n(c),\,c\ne 0$ (cf. [2, 16]).

The spaces $M$ stated in Theorem 4 can be naturally immersed into $\hat{M}_{n+p}(c)$ with higher codimension via the standard holomorphic immersion of $\hat{M}_n(c)$ into $\hat{M}_{n+p}(c)$ as follows

$$\begin{aligned} M\longrightarrow \hat{M}_n(c)\longrightarrow \hat{M}_{n+p}(c). \end{aligned}$$

Clearly, such an immersion is not full. Next, we shall discuss an example of CR-submanifolds with maximal CR-dimension in $\mathbb{C }P_q$, which are irreducible to real hypersurfaces in a totally geodesic holomorphic submanifold of $\mathbb{C }P_q$.

We denote by $(z_0:z_1:\cdots :z_n)$ the homogeneous coordinates of $\mathbb{C }P_n$. Then, the Segre embedding $S_{m,l}:\mathbb{C }P_m\times \mathbb{C }P_l\rightarrow \mathbb{C }P_{m+l+ml}$ is given by

$$\begin{aligned} S_{m,l}(z,w)=(z_0w_0:\cdots :z_0w_l:z_1w_0:\cdots :z_1w_l:\cdots :z_mw_0:\cdots z_mw_l) \end{aligned}$$

where $(z_0:z_1:\cdots :z_m)\in \mathbb{C }P_m$ and $(w_0:w_1:\cdots :w_l)\in \mathbb{C }P_l$.

Example 4

(The standard CR-products $\mathbb{C }P_{n-1}\times \mathbb{R }P^1$) We consider $\mathbb{R }P^l$ as a totally geodesic, totally real submanifold in $\mathbb{C }P_l$, and $\mathbb{C }P_{m+l+ml}$ as a totally geodesic, holomorphic submanifold in $\mathbb{C }P_q,\,m+l+ml\le q$. The standard CR-product $\mathbb{C }P_{m}\times \mathbb{R }P^l$ can be immersed into $\mathbb{C }P_{q}$ via $S_{m,l}$ as follows: (cf. [3])

$$\begin{aligned} \mathbb{C }P_{m}\times \mathbb{R }P^l\longrightarrow \mathbb{C }P_{m}\times \mathbb{C }P_l \stackrel{S_{m,l}}{\longrightarrow }\mathbb{C }P_{m+l+ml}\longrightarrow \mathbb{C }P_q. \end{aligned}$$

(18)

In particular, $\mathbb{C }P_{n-1}\times \mathbb{R }P^1$ is a CR-submanifold of maximal CR-dimension in $\mathbb{C }P_q, 2n-1\le q$.

Theorem 5

([3]) Let $M$ be a CR-product in $\mathbb{C }P_q,\,\dim _{\mathbb{C }}{\fancyscript{D}}=m$ and $\dim _{\mathbb{R }}{\fancyscript{D}}^\perp =l$. Then, we have

$$\begin{aligned} ||h||^2\ge 4ml \end{aligned}$$

and equality holds if and only if $M$ is given by the immersion (18).

Theorem 6

Let $M=\mathbb{C }P_{n-1}\times \mathbb{R }P^1$. Then, $M$ is a cyclic parallel CR-submanifold of maximal CR-dimension in $\mathbb{C }P_{q}$.

Proof

Since $\mathbb{C }P_{n-1}$ and $\mathbb{R }P^1$ are leaves of ${\fancyscript{D}}$ and ${\fancyscript{D}}^\perp$, respectively, and they are totally geodesic in $\mathbb{C }P_{q}$, by the Gauss formula, we see that

$$\begin{aligned} h({\fancyscript{D}},{\fancyscript{D}})=0; \qquad h(\xi ,\xi )=0. \end{aligned}$$

Further, as $M$ is a Riemannian product $\mathbb{C }P_{n-1}\times \mathbb{R }P^1$, both distributions ${\fancyscript{D}}$ and ${\fancyscript{D}}^\perp$ are auto-parallel, that is,

$$\begin{aligned} \nabla :\varGamma ({\fancyscript{D}})\rightarrow \Omega ^1(M)\,_{\Omega ^0(M)}\otimes \varGamma ({\fancyscript{D}});\qquad \nabla :\varGamma ({\fancyscript{D}}^\perp )\rightarrow \Omega ^1(M)\,_{\Omega ^0(M)}\otimes \varGamma ({\fancyscript{D}}^\perp ). \end{aligned}$$

Therefore, we have

$$\begin{aligned} (\nabla _Xh)(Y,Z)&= \nabla ^\perp _X h(Y,Z)-h(\nabla _XY,Z)-h(Y,\nabla _XZ)=0 \\ (\nabla _Xh)(\xi ,\xi )&= \nabla ^\perp _X h(\xi ,\xi )-2h(\nabla _X\xi ,\xi )=0 \end{aligned}$$

for any $X\in \varGamma (TM)$ and $Y,Z\in \varGamma ({\fancyscript{D}})$. By using the above two equations and the Codazzi equation, we have

$$\begin{aligned} (\nabla _\xi h)(Y,\xi )&= (\nabla _Y h)(\xi ,\xi )=0 \\ (\nabla _Xh)(Y,\xi )&= (\nabla _\xi h)(X,Y)-c\langle \phi X,Y\rangle N=-c\langle \phi X,Y\rangle N \end{aligned}$$

for any $X, Y\in \varGamma ({\fancyscript{D}})$. Hence, $M$ satisfies (17) and so it is cyclic parallel. $\square$

Remark 4

By using a similar manner as in the above proof, we may verify that such standard CR-products with higher CR-codimension are also cyclic parallel.

5 Lemmas

Throughout this section, suppose $M$ is a $(2n-1)$-dimensional CR-submanifold of maximal CR-dimension in $\hat{M}_{n+p}(c),\,c\ne 0,\,n\ge 2$ and $M$ is cyclic parallel or equivalent, it satisfies (17), that is,

$$\begin{aligned} (\nabla _Xh)(Y,Z)=-c\{\eta (Y)\langle \phi X,Z\rangle +\eta (Z)\langle \phi X,Y\rangle \}N \end{aligned}$$

for any $X, Y, and Z\in \varGamma (TM)$. By (15), we can see that the condition (17) is equivalent to

$$\begin{aligned} (\nabla _XA)_\sigma Y=-c\langle \sigma ,N\rangle \{\eta (Y)\phi X+\langle \phi X,Y\rangle \xi \} \end{aligned}$$

(19)

for any $X, Y\in \varGamma (TM)$ and $\sigma \in \varGamma (\nu )$.

Lemma 4

 

• (a) $CH=0$;

• (b) $\langle H,N\rangle Ch=0$.

Proof

Note that the Eq. (17) implies that $\nabla ^\perp H=0$, and by the Ricci equation (14), we have

$$\begin{aligned} -2c\langle \phi Y,Z\rangle CH+h(Y,A_H Z)-h(Z,A_H Y)=0 \end{aligned}$$

for any $Y,Z\in \varGamma (TM)$. By differentiating this equation covariantly in the direction of $X\in \varGamma (TM)$, we have

$$\begin{aligned}&-2c\langle (\nabla _X\phi ) Y,Z\rangle CH-2c\langle \phi Y,Z\rangle (\nabla _XC)H+(\nabla _Xh)(Y,A_HZ)+h(Y,(\nabla _XA)_HZ)\\&-(\nabla _Xh)(Z,A_HY)-h(Z,(\nabla _XA)_HY)=0. \end{aligned}$$

By using (9)–(12) and (17), we have

$$\begin{aligned}&2\{-\eta (Y)\langle A_NX,Z\rangle +\eta (Z)\langle A_NX,Y\rangle \} CH +2\langle \phi Y,Z\rangle \{\eta (A_HX)N \nonumber \\&-\langle H,N\rangle h(X,\xi )\} +\{\eta (Z)\langle \phi X,A_HY\rangle +\eta (A_HY)\langle \phi X,Z\rangle \nonumber \\&-\eta (Y)\langle \phi X,A_HZ\rangle -\eta (A_HZ)\langle \phi X,Y\rangle \}N +\langle H,N\rangle \{\eta (Y)h(Z,\phi X) \nonumber \\&+\langle \phi X,Y\rangle h(Z,\xi ) -\eta (Z)h(Y,\phi X) -\langle \phi X,Z\rangle h(Y,\xi )\}=0. \end{aligned}$$

(20)

If we substitute $X=\xi ,\,Y\in \varGamma ({\fancyscript{D}})$ and $Z=\phi Y$ in the above equation, then $\langle h(\xi ,\xi ),H\rangle N-\langle H,N\rangle h(\xi ,\xi )=0$ and hence $CH=0$.

Furthermore, after putting $Y=X\in \varGamma ({\fancyscript{D}})$ and $Z=\phi X$ in (20), we get

$$\begin{aligned} \langle H,N\rangle Ch(X,\xi )=0 \end{aligned}$$

for any $X\in \varGamma ({\fancyscript{D}})$. Next, by putting $Z=\xi$ in (20) and making use of the above equation, we obtain

$$\begin{aligned} \langle H,N\rangle Ch(Y,\phi X)=0 \end{aligned}$$

for any $X, Y\in \varGamma (TM)$. By these two equations and the fact that $Ch(\xi ,\xi )=0 (=CH)$, we obtain Statement (b). $\square$

Lemma 5

For any $X\in \varGamma (TM)$,

• (a) $\langle \phi A_N\xi ,X\rangle N=-h(\xi ,\phi X)+Ch(\xi ,X)$;

• (b) $d\alpha (X)=2\eta (A_N\phi A_NX)$;

• (c) $2Ch(A_NX,\xi )=\alpha Ch(X,\xi )$.

Proof

Statement (a) can be obtained easily from $X=\xi$ in (11) and Lemma 4.

Taking into account that $CH=0$ again, we see that $h(\xi ,\xi )=\alpha N$. It follows from (17), (10), and Statement (a) that

$$\begin{aligned} 0=(\nabla _Xh)(\xi ,\xi )=d\alpha (X)N+\alpha Ch(X,\xi )+2\langle \phi A_N\xi ,A_NX\rangle N-2Ch(\xi ,A_NX) \end{aligned}$$

for any $X\in \varGamma (TM)$. Statements (b) and (c) are the $J{\fancyscript{D}}^\perp$- and $\nu$-component of this equation, respectively. $\square$

Lemma 6

For any $X, Y, Z, and W\in \varGamma (TM)$,

$$\begin{aligned}&c\{ -\langle \phi Y,\phi Z\rangle \langle A_NX,W\rangle +\langle \phi X,\phi Z\rangle \langle A_NY,W\rangle \nonumber \\&\quad -\langle \phi Y,\phi W\rangle \langle A_NX,Z\rangle +\langle \phi X,\phi W\rangle \langle A_NY,Z\rangle \nonumber \\&\quad +\langle \phi Y,Z\rangle \langle (\phi A_N-A_N\phi )X,W\rangle -\langle \phi X,Z\rangle \langle (\phi A_N-A_N\phi )Y,W\rangle \nonumber \\&\quad +\langle \phi Y,W\rangle \langle (\phi A_N-A_N\phi )X,Z\rangle -\langle \phi X,W\rangle \langle (\phi A_N-A_N\phi )Y,Z\rangle \nonumber \\&\quad -2\langle \phi X,Y\rangle \langle (\phi A_N-A_N\phi )Z,W\rangle \} \nonumber \\&\quad -\langle h(Y,Z),h(X,A_NW)\rangle +\langle h(X,Z),h(Y,A_NW)\rangle \nonumber \\&\quad -\langle h(Y,W),h(X,A_NZ)\rangle +\langle h(X,W),h(Y,A_NZ)\rangle \nonumber \\&\quad -\langle h(Z,W),h(X,A_NY)\rangle +\langle h(Z,W),h(Y,A_NX)\rangle =0,\end{aligned}$$

(21)

$$\begin{aligned}&\quad c\big \{\! -\langle Y,Z\rangle \langle A_\sigma X,W\rangle +\langle X,Z\rangle \langle A_\sigma Y,W\rangle \nonumber \\&\quad -\langle Y,W\rangle \langle A_\sigma X,Z\rangle +\langle X,W\rangle \langle A_\sigma Y,Z\rangle \nonumber \\&\quad -\langle \phi Y,Z\rangle \langle \phi A_\sigma \phi X,\phi W\rangle +\langle \phi X,Z\rangle \langle \phi A_\sigma \phi Y,\phi W\rangle \nonumber \\&\quad -\langle \phi Y,W\rangle \langle \phi A_\sigma \phi X,\phi Z\rangle +\langle \phi X,W\rangle \langle \phi A_\sigma \phi Y,\phi Z\rangle \nonumber \\&\quad -2\langle \phi X,Y\rangle \{\eta (Z)\langle A_\sigma \xi ,\phi W\rangle +\eta (W)\langle A_\sigma \xi ,\phi Z\rangle +\langle A_{C\sigma }Z,W\rangle \}\big \} \nonumber \\&\quad -\langle h(Y,Z),h(X,A_\sigma W)\rangle +\langle h(X,Z),h(Y,A_\sigma W)\rangle \nonumber \\&\quad -\langle h(Y,W),h(X,A_\sigma Z)\rangle +\langle h(X,W),h(Y,A_\sigma Z)\rangle \nonumber \\&\quad -\langle h(Z,W),h(X,A_\sigma Y)\rangle +\langle h(Z,W),h(Y,A_\sigma X)\rangle =0. \end{aligned}$$

(22)

proof

Differentiating the following equation

$$\begin{aligned} (\nabla _Yh)(Z,W)=-c\{\eta (Z)\langle \phi Y,W\rangle +\eta (W)\langle \phi Y,Z\rangle \}N \end{aligned}$$

covariantly in the direction of $X\in \varGamma (TM)$, with the help of (9) and (10), we have

$$\begin{aligned} (\nabla ^2_{XY} h)(Z,W)&= -c\{ \langle \phi A_NX,Z\rangle \langle \phi Y,W\rangle +\eta (Z)\eta (Y)\langle A_NX,W\rangle \\&+ \langle \phi A_NX,W\rangle \langle \phi Y,Z\rangle +\eta (W)\eta (Y)\langle A_NX,Z\rangle \\&- 2\eta (Z)\eta (W)\langle A_NX,Y\rangle \}N \\&-c\{ \eta (Z)\langle \phi Y,W\rangle +\eta (W)\langle \phi Y,Z\rangle \}Ch(X,\xi ). \end{aligned}$$

It follows from (8), (6), (14), and this equation that

$$\begin{aligned}&c\big \{\{ -\langle \phi A_NX,Z\rangle \langle \phi Y,W\rangle -\eta (Z)\eta (Y)\langle A_NX,W\rangle \nonumber \\&\quad -\langle \phi A_NX,W\rangle \langle \phi Y,Z\rangle -\eta (W)\eta (Y)\langle A_NX,Z\rangle \nonumber \\&\quad +\langle \phi A_NY,Z\rangle \langle \phi X,W\rangle +\eta (Z)\eta (X)\langle A_NY,W\rangle \nonumber \\&\quad +\langle \phi A_NY,W\rangle \langle \phi X,Z\rangle +\eta (W)\eta (X)\langle A_NY,Z\rangle \}N \nonumber \\&\quad -\{\eta (Z)\langle \phi Y,W\rangle +\eta (W)\langle \phi Y,Z\rangle \}Ch(X,\xi ) \nonumber \\&\quad +\{\eta (Z)\langle \phi X,W\rangle +\eta (W)\langle \phi X,Z\rangle \}Ch(Y,\xi )\big \}\nonumber \\&\quad =c\big \{\!-\langle Y,Z\rangle h(X,W)+\langle X,Z\rangle h(Y,W) \nonumber \\&\quad -\langle \phi Y,Z\rangle h(\phi X,W)+\langle \phi X,Z\rangle h(\phi Y,W) \nonumber \\&\quad -\langle Y,W\rangle h(X,Z)+\langle X,W\rangle h(Y,Z) \nonumber \\&\quad -\langle \phi Y,W\rangle h(\phi X,Z)+\langle \phi X,W\rangle h(\phi Y,Z) \nonumber \\&\quad +2\langle \phi X,Y\rangle \{h(\phi Z,W)+h(\phi W,Z)-Ch(Z,W)\} \big \} \nonumber \\&\quad -h(A_{h(Y,Z)}X,W)+h(A_{h(X,Z)}Y,W) \nonumber \\&\quad -h(A_{h(Y,W)}X,Z)+h(A_{h(X,W)}Y,Z) \nonumber \\&\quad -h(A_{h(Z,W)}X,Y)+h(A_{h(Z,W)}Y,X). \end{aligned}$$

(23)

The Eq. (21) is the $J{\fancyscript{D}}^\perp$-component of this equation. Next, it follows from Lemma 5(a) that

$$\begin{aligned} \langle h(Z,\phi Y)-\eta (Z)Ch(\xi ,Y),\sigma \rangle =-\langle h(\phi ^2 Z,\phi Y),\sigma \rangle =\langle \phi A_\sigma \phi Y,\phi Z\rangle \end{aligned}$$

for any $Y,Z\in \varGamma (TM)$ and $\sigma \in \varGamma (\nu )$. With the help of this equation, after taking inner product of both sides of (23) with $\sigma \in \varGamma (\nu )$, we obtain (22). $\square$

Lemma 7

$A_N\xi =\alpha \xi$.

Proof

Suppose that $\beta =||\phi A_N\xi ||>0$ at some point $x\in M$. Then, we can write

$$\begin{aligned} A_N\xi =\alpha \xi +\beta U \end{aligned}$$

(24)

where $U=-\beta ^{-1}\phi ^2A_N\xi$ and hence from Lemma 5(c), we have

$$\begin{aligned} Ch(\xi ,U)=0. \end{aligned}$$

(25)

Next, by substituting $Z=W=\xi$ in (21), we obtain

$$\begin{aligned} \langle h(X,U),h(Y,\xi )\rangle -\langle h(Y,U),h(X,\xi )\rangle =0 \end{aligned}$$

(26)

for any $X, Y\in T_xM$. By putting $Y=\xi$ in this equation, with the help of (25), we obtain $\alpha A_NU-\beta A_N\xi =0$ and so

$$\begin{aligned} A_NU=\beta \xi +\gamma U, \quad (\alpha \gamma =\beta ^2). \end{aligned}$$

(27)

Hence, from (24) and (27), we have

$$\begin{aligned} A_N^2\xi =(\alpha ^2+\beta ^2)\xi +\beta (\alpha +\gamma )U \end{aligned}$$

(28)

$$\begin{aligned} A_N^2U=\beta (\alpha +\gamma )\xi +(\beta ^2+\gamma ^2)U. \end{aligned}$$

(29)

On the other hand, by putting $X=U$ in (26) and using (25) and (27), we have $A_{h(U,U)}\xi -\beta A_NU=0$ or

$$\begin{aligned} A_{C^2h(U,U)}\xi =0. \end{aligned}$$

(30)

Finally, with the help of (24), (25), (27)–(30), Lemma 5(c) and the fact that $h(\xi ,\xi )=\alpha N$, after substituting $X=W=U,\,Y=Z=\xi$ in (21), gives

$$\begin{aligned} 0=c\alpha -\alpha \langle A_NU,A_NU\rangle +\gamma \langle A_N\xi ,A_N\xi \rangle =c\alpha . \end{aligned}$$

But from (27), $\alpha \gamma =\beta ^2>0$. This is a contradiction. Accordingly, $A_N\xi =\alpha \xi$ at each point of $M$. $\square$

Lemma 8

 

• (a) $\alpha$ is a constant;

• (b) $(A_N\phi A_N-\alpha \phi A_N-c\phi )X+A_{h(\phi X,\xi )}\xi =0$, for any $X\in \varGamma (TM)$;

• (c) $\alpha (\phi A_N-A_N\phi )=0$.

Proof

Statement (a) is directly from Lemma 5(b) and Lemma 7. Next, from Lemma 7, we have

$$\begin{aligned} \langle h(Y,\xi ),N\rangle =\alpha \eta (Y) \end{aligned}$$

for any $Y\in \varGamma (TM)$. It follows from this equation that

$$\begin{aligned} \langle (\nabla _Xh)(Y,\xi )+h(Y,\nabla _X\xi ),N\rangle +\langle h(Y,\xi ),\nabla _X^\perp N\rangle = d\alpha (X)\eta (Y)+\alpha \langle \nabla _X\xi ,Y\rangle . \end{aligned}$$

By applying (10), (17), Lemma 5(a), and Lemma 8(a), this equation becomes

$$\begin{aligned} \langle (A_N\phi A_N-\alpha \phi A_N-c\phi )X,Y\rangle +\langle h(\phi X,\xi ),h(Y,\xi )\rangle =0 \end{aligned}$$

(31)

for any $X, Y\in \varGamma (TM)$ and so we obtain Statement (b). Finally, by letting $X=Y$ in (31), we have $\alpha \langle \phi A_NX,X\rangle =0$, for any $X\in \varGamma (TM)$, this deduces Statement (c). $\square$

Lemma 9

For any $X, Y\in \varGamma ({\fancyscript{D}})$ and $\sigma \in \varGamma (\nu )$,

$$\begin{aligned} 2\langle h(\phi X,\xi ),h(Y,A_\sigma \xi )\rangle +\langle 2A_NY-\alpha Y,A_\sigma \phi A_NX\rangle =0. \end{aligned}$$

Proof

By using Lemma 5(c) and Lemma 7, we have

$$\begin{aligned} 2h(A_NY,\xi )=2\langle h(A_NY,\xi ),N\rangle N-2C^2h(A_NY,\xi ) =\alpha ^2\eta (Y)N+\alpha h(Y,\xi ) \end{aligned}$$

for any $Y\in \varGamma (TM)$. By differentiating this equation covariantly in the direction of $X\in \varGamma (TM)$, we have

$$\begin{aligned}&2\{(\nabla _Xh)(A_NY,\xi ) +h((\nabla _XA)_NY+A_{\nabla ^\perp _X N}Y,\xi )+h(A_NY,\nabla _X\xi )\}\\&=\alpha ^2\{\langle \nabla _X\xi ,Y\rangle N+\eta (Y)\nabla ^\perp _X N\} +\alpha \{(\nabla _Xh)(Y,\xi )+ h(Y,\nabla _X\xi )\}. \end{aligned}$$

By using (10), (17), Lemma 5(a), and Lemma 8(a), this equation becomes

$$\begin{aligned}&\!-\!2c\{\langle \phi X,A_NY\rangle N\!+\!\eta (Y)h(\phi X,\xi )\!+\!\alpha \langle \phi X,Y\rangle N\} \!+\!2\{h(A_{h(\phi X,\xi )}Y,\xi )\!+\!h(A_NY,\phi A_NX)\} \\&\quad \!=\!\alpha ^2\{\langle \phi A_NX,Y\rangle N\!+\!\eta (Y)h(\phi X,\xi )\} \!+\!\alpha \{\!-\!c\langle \phi X,Y\rangle N\!+\! h(Y,\phi A_NX)\}. \end{aligned}$$

By first, putting $X, Y\in \varGamma ({\fancyscript{D}})$ and then taking inner product of both sides of this equation with $\sigma \in \varGamma (\nu )$, we obtain the lemma. $\square$

6 Proof of Theorem 1

We shall consider two cases: (I) $M$ is mixed totally geodesic and (II) $M$ is non-mixed totally geodesic.

• Case (I) $M$ is mixed totally geodesic. By Lemma 4(a) and Lemma 7, we have

  $$\begin{aligned} h(Y,\xi )=\eta (Y)h(\xi ,\xi )=\alpha \eta (Y)N \end{aligned}$$

  (32)

  for any $Y\in \varGamma (TM)$. It follows from (10) that $\nabla ^\perp N=0$. Moreover, by applying (10), (17), and (32), we obtain

  $$\begin{aligned}&0=\langle (\nabla _Xh)(Y,\xi ),\sigma \rangle =\langle \nabla ^\perp _Xh(Y,\xi ),\sigma \rangle -\langle h(Y,\nabla _X\xi ),\sigma \rangle \\&\quad =\langle h(Y,\phi A_NX),\sigma \rangle \nonumber \end{aligned}$$

  for any $X, Y\in \varGamma (TM)$ and $\sigma \in \varGamma (\nu )$. This means that

  $$\begin{aligned} A_\sigma \phi A_N=0 \end{aligned}$$

  (33)

  for any $\sigma \in \varGamma (\nu )$. On the other hand, by Lemma 8(b), we have

  $$\begin{aligned} A_N\phi A_N-\alpha \phi A_N-c\phi =0. \end{aligned}$$

  As $c\ne 0$, we can observe from the above equation that $A_N|_{{\fancyscript{D}}}$ is a vector bundle automorphism on ${\fancyscript{D}}$. Hence, for any $\sigma \in \varGamma (\nu )$, we have $A_\sigma |_{{\fancyscript{D}}} =0$ by (33). Also, we have $A_\sigma \xi =0$ by using Lemma 4(a). We conclude that $A_\sigma =0$ for any $\sigma \in \varGamma (\nu )$. Further, since $A_N\ne 0,\,\nu _x$ is the $J$-invariant orthogonal complementary subspace of the first normal space in $T_xM^\perp$, at each $x\in M$. Also, since $\nabla ^\perp N=0,\,\nu$ is a parallel normal subbundle of $TM^\perp$. By applying Theorem 3, $M$ is contained in a totally geodesic holomorphic submanifold $\hat{M}_n(c)$ of $\hat{M}_{n+p}(c)$ as a real hypersurface. We denote by $N^{\prime }$ a unit normal vector field, $\nabla ^{\prime }$, the Levi-Civita connection, $A^{\prime }$ the shape operator of $M$, immersed in $\hat{M}_n(c)$. Further, let $(\phi ^{\prime },\xi ^{\prime },\eta ^{\prime })$ denote the almost contact structure on $M$ induced by complex structure of $\hat{M}_n(c)$. Since $\hat{M}_n(c)$ is totally geodesic in $\hat{M}_{n+p}(c)$ and $Ch=0$, we can see that $\nabla ^{\prime }_XY=\nabla _XY,\,A^{\prime }=A_N,\,\phi ^{\prime }=\phi ,\,\eta ^{\prime }=\eta ,\,\xi ^{\prime }=\xi$, and $N^{\prime }=N$. Then, by (19), we have

  $$\begin{aligned} (\nabla ^{\prime }_XA^{\prime })Y&= (\nabla _XA)_NY=-c\{\eta (Y))\phi X+\langle \phi X,Y\rangle \xi \} \\&= -c\{\eta ^{\prime }(Y))\phi ^{\prime } X+\langle \phi ^{\prime } X,Y\rangle \xi ^{\prime }\} \end{aligned}$$

  for any vectors $X, Y$ tangent $M$. By using Theorem 4, we obtain Case (a) and Case (b)(i) and (ii) in Theorem 1.

• Case (II) $M$ is non-mixed totally geodesic. Let $x\in M$, and $X\in {\fancyscript{D}}_x$ be a unit vector with $A_NX=\lambda X$. If $h(X,\xi )=0$, then we also have $h(\phi X,\xi )=Ch(X,\xi )=0$ and

  $$\begin{aligned} \lambda A_N\phi X-(\alpha \lambda +c)\phi X=0, \qquad (\text{ by} \text{ Lemma} \text{8(b)}). \end{aligned}$$

  If $\alpha =0$, then $\lambda \ne 0$ and $A_N\phi X=c\lambda ^{-1}\phi X$. On the other hand, if $\alpha \ne 0$, then by Lemma 8(c), $A_N\phi X=\lambda X$. From these observations, there is an integer $m\ge 1$ and we may choose an orthonormal basis of ${\fancyscript{D}}_x$ formed by eigenvectors $E_1, E_2=\phi E_1,\ldots ,E_{2n-1}, E_{2n-2}=\phi E_{2n-1}$ of $A_N$ such that

  $$\begin{aligned} h(E_i,\xi )&\ne 0, \quad (1\le i\le 2m) \end{aligned}$$

  (34)
  $$\begin{aligned} h(E_a,\xi )&= 0, \quad (2m+1\le a\le 2n-2). \end{aligned}$$

  (35)

  In the rest of this section, we use the following convention of indices:

  $$\begin{aligned} i,j,\ldots&\in \{1,2,\ldots ,2m\};\\ a,b,\ldots&\in \{2m+1,\ldots ,2n-2\}. \end{aligned}$$

  For simplicity, we write $\sigma _i=h(E_i,\xi )$ and $A_i=A_{\sigma _i}$. It follows from Lemma 5(c) and Lemma 8(b) that

  $$\begin{aligned} A_NE_i&= \frac{\alpha }{2}E_i \end{aligned}$$

  (36)
  $$\begin{aligned} A_i\xi&= \frac{\alpha ^2+4c}{4} E_i \end{aligned}$$

  (37)
  $$\begin{aligned} \langle \sigma _i,h(X,\xi )\rangle&= \frac{\alpha ^2+4c}{4}\langle E_i,X\rangle \end{aligned}$$

  (38)

  for any $X\in T_xM$. We can further observe from (38) that

  $$\begin{aligned} ||\sigma _i||^2=\frac{\alpha ^2+4c}{4}>0. \end{aligned}$$

  (39)

  By using (36)–(39), after putting $X=\phi E_i,\,Y=E_j$, and $\sigma =\sigma _k$ in Lemma 9, we obtain $(\alpha ^2+4c)\langle \sigma _i,h(E_j,E_k)\rangle =0$, and so

  $$\begin{aligned} \langle A_{i}E_j,E_k\rangle =0. \end{aligned}$$

  (40)

  Now, we wish to prove that

  $$\begin{aligned} A_{i}E_j=\frac{\alpha ^2+4c}{4} \delta _{ij}\xi . \end{aligned}$$

  (41)

  If $m=n-1$, then (37) and (40) imply (41). Next, suppose $m<n-1$. Then, by letting $Y=Z=\xi ,\,X=E_j,\,W=E_a$, and $\sigma =\sigma _i$ in (22), with the help of (35)–(37), we have $c\langle A_{i}E_j,E_a\rangle =0$, that is,

  $$\begin{aligned} \langle A_{i}E_j,E_a\rangle =0. \end{aligned}$$

  From the above equation, (37) and (40), we also obtain (41). By putting $X=\xi ,\,Y=E_i,\,Z=E_j,\,W=E_k$, and $\sigma =\sigma _l$ in (22), we have

  $$\begin{aligned} \frac{\alpha ^2}{4}\{\delta _{jk}\delta _{il}+\delta _{ji}\delta _{kl}+\delta _{ki}\delta _{jl}\} =\langle Ch(E_j,E_k),Ch(E_i,E_l)\rangle . \end{aligned}$$

  (42)

  If we first put $E_i=E_j=E_k=E_l$, and next follow by $E_j=E_i,\,E_k=E_l=\phi E_i$ in the above equation, then

  $$\begin{aligned} \frac{3\alpha ^2}{4}&= \langle Ch(E_i,E_i),Ch(E_i,E_i)\rangle \\ \frac{\alpha ^2}{4}&= \langle Ch(E_i,\phi E_i),Ch(E_i,\phi E_i)\rangle =\langle Ch(E_i,E_i),Ch(E_i,E_i)\rangle . \\ \end{aligned}$$

  These three equations, together with (36) and (39), give

  $$\begin{aligned}&\alpha =0 \end{aligned}$$

  (43)
  $$\begin{aligned}&c >0; \quad \text{(without} \text{ loss} \text{ of} \text{ generality,} \text{ we} \text{ assume} {c=1)} \end{aligned}$$

  (44)
  $$\begin{aligned}&h(E_i,E_j) =0 . \end{aligned}$$

  (45)

Lemma 10

Suppose $m<n-1$ and let $A_NE_a=\lambda _a E_a$. Then,

• (a) $Ch(E_a,E_b)=0$,

• (b) $\lambda _a\in \{1,-1\}$,

• (c) $\phi A_N-A_N\phi =0$.

Proof

From (31), (35), (43), and (44), we have $\lambda _a\ne 0$ and $A_N\phi E_a=\lambda _a^{-1}\phi E_a$. Hence, after putting $X=\phi E_a$ and $Y=E_b$ in Lemma 9, we obtain Statement (a). Furthermore, by putting $X=W=E_i$ and $Y=Z=E_a$, and $X=E_i,\,Y=E_a,\,Z=\phi E_a$, and $W=\phi E_i$, respectively, in (21), we have

$$\begin{aligned} 0&= -\lambda _a+2\lambda _a\langle h(E_i,E_a),h(E_i,E_a)\rangle \\ 0&= \lambda _a-\lambda _a^{-1}+ \lambda _a^{-1} \langle h(E_i,\phi E_a),h(\phi E_i,E_a)\rangle + \lambda _a \langle h(E_i,E_a),h(\phi E_i,\phi E_a)\rangle \\&= \{\lambda _a-\lambda _a^{-1}\}\{1-\langle h(E_i,E_a),h(E_i,E_a)\}. \end{aligned}$$

These two equations imply that $\lambda _a=\lambda _a^{-1}$. Hence, we obtain Statement (b) and (c) as $AE_i=A\phi E_i=0$. $\square$

Now, we consider two subcases: $||A_N||=0$ and $||A_N||\ne 0$.

Subcase (II-a) $||A_N||=0$.

In this case, we have $m=n-1$ at each $x\in M$ by (36), (43), and Lemma 10(b). From Lemma 7 and (45), we see that $\langle h(X,Y),N\rangle =0$, for any $X\in \varGamma ({\fancyscript{D}})$ and $Y\in \varGamma (TM)$. Hence, $M$ is a CR-product by Lemma 2. Furthermore, it follows from (38), (45), and $h(\xi ,\xi )=0$ that $||h||^2=2(2n-2)$. According to Theorem 5, $M$ is an open part of the standard CR-product $\mathbb{C }P_{n-1}\times \mathbb{R }P^1$, and we obtain Case (b)(iii) in Theorem 1.

Subcase (II-b) $||A_N||\ne 0$.

From Lemma 4(b), we have $\text{ Trace}( A_N|_{{\fancyscript{D}}_x})=\langle H,N\rangle =0$. By using (36), (43), Lemma 10(b), and the continuity of the eigenvalue functions, we can see that $m<n-1$ and $A_N$ has three distinct constant eigenvalues $0,\,1$, and $-1$ with multiplicities $2m,\,n-m-1$, and $n-m-1$, respectively, at each $x\in M$.

For $\lambda \in \{0,1,-1\}$, we denote by $\fancyscript{T}_{\lambda }$ the subbundle of ${\fancyscript{D}}$ foliated by eigenspace of $A_N|_{{\fancyscript{D}}}$ corresponding to $\lambda$. From Lemma 10(c), we see that each $\fancyscript{T}_{\lambda }$ is $\phi$-invariant. We shall show that $\fancyscript{T}_0$ is auto-parallel, that is,

$$\begin{aligned} \varGamma (\fancyscript{T}_0)\stackrel{\nabla }{\longrightarrow }\Omega ^1(M)\otimes \varGamma (\fancyscript{T}_0). \end{aligned}$$

For any $X\in \varGamma (TM)$ and $Y\in \varGamma (\fancyscript{T}_0)$, we have

$$\begin{aligned} \langle \nabla _XY,\xi \rangle =-\langle Y,\phi A_NX\rangle =0. \end{aligned}$$

Next, from (17), we have

$$\begin{aligned} -\langle \phi X,Y\rangle \xi =(\nabla _XA)_NY=-A_N\nabla _XY-A_{Ch(X,\xi )}Y =-A_N\nabla _XY-A_{h(\phi X,\xi )}Y. \end{aligned}$$

If $X\in \varGamma (\fancyscript{T}_1\oplus \fancyscript{T}_{-1}\oplus \text{ Span}\{\xi \})$, it clearly that $A_N\nabla _XY=0$; if $X\in \varGamma (\fancyscript{T}_0)$, then by (37) and the above equation, we have $A_N\nabla _XY=0$ too. From these observations, we have $\nabla _XY\in \varGamma (\fancyscript{T}_{0})$, for any $X\in \varGamma (TM)$ and $Y\in \varGamma (\fancyscript{T}_{0}).$

For any $X\in \varGamma (\fancyscript{T}_0)$ and $Y,Z\in \varGamma (\fancyscript{T}_1\oplus \fancyscript{T}_{-1}\oplus \text{ Span}\{\xi \})$, from Lemma 10(a), we see that $h(Y,Z)=\langle A_NY,Z\rangle N$. It follows that

$$\begin{aligned} (\nabla _Xh)(Y,Z)&=\nabla ^\perp _Xh(Y,Z)-h(\nabla _XY,Z)-h(Y,\nabla _XZ) \\&=\{X\langle A_NY,Z\rangle -\langle A_N\nabla _XY,Z\rangle -\langle A_NY,\nabla _XZ\rangle \}N \\&\quad -\langle A_NY,Z\rangle Ch(X,\xi ). \end{aligned}$$

In particular, if we choose $Y=Z\in \varGamma (\fancyscript{T}_1)$ with $||Y||=1$, then

$$\begin{aligned} C(\nabla _Xh)(Y,Z)=h(X,\xi )\ne 0. \end{aligned}$$

This is a contradiction, so this case cannot occur.

Conversely, all these submanifolds satisfy the condition (17) as we have discussed in Sect. 4. This completes the proof.

7 Proof Theorem 2

Suppose $M$ is a $(2n-1)$-dimensional CR-submanifold of maximal CR-dimension in $\hat{M}_{n+p}(c),\,c\ne 0,\,n\ge 2$. We define a tensor field $T$ on $M$ by

$$\begin{aligned} T(X,Y,Z)=(\nabla _Xh)(Y,Z)+c\{\eta (Y)\langle \phi X,Z\rangle +\eta (Z)\langle \phi X,Y\rangle \}N \end{aligned}$$

for any $X, Y, and Z\in \varGamma (TM)$. Let $e_1,e_2,\ldots ,e_{2n-1}$ be a local field of orthonormal vectors in $\varGamma (TM)$. Then,

$$\begin{aligned} ||T||^2=||\nabla h||^2+4(n-1)c^2+4c\sum ^{2n-1}_{j=1}\langle (\nabla _{e_j}h)(\xi ,\phi e_j),N\rangle . \end{aligned}$$

On the other hand, by the Codazzi equation, we have

$$\begin{aligned} \sum ^{2n-1}_{j=1}\langle (\nabla _{e_j}h)(\xi ,\phi e_j),N\rangle =\sum ^{2n-1}_{j=1}\langle (\nabla _{\xi }h)(e_j,\phi e_j),N\rangle -2(n-1)c=-2(n-1)c. \end{aligned}$$

Combining these two equations, we have

$$\begin{aligned} 0\le ||T||^2=||\nabla h||^2-4(n-1)c^2 \end{aligned}$$

and equality holds if and only if $M$ satisfies (17). By Theorem 1, we obtain the theorem.