The Chern-Moser-Tanaka Invariant on Pseudo-Hermitian Almost CR Manifolds

Abstract

We study on the Chern-Moser-Tanaka invariant (Chern, Acta Math 133:219–271, 1974, [5], Tanaka, Japan J Math 12:131–190, 1976, [14]) of pseudo-conformal transformations on pseudo-Hermitian almost CR manifolds.

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1 Introduction

A contact manifold $(M,\eta )$ admits the fundamental structures which enrich the geometry. One is a Riemannian metric g compatible to the contact form $\eta$ and we obtain a contact Riemannian manifold $(M;\eta ,g)$. The other is a pseudo-Hermitian and strictly pseudo-convex structure $(\eta , L)$ (or $(\eta ,J)$), where L is the Levi form associated with an endomorphism J on D (= kernel of $\eta$) such that $J^2=-I$. $(M;\eta ,J)$ is called a strictly pseudo-convex, pseudo-Hermitian manifold (or almost CR manifold). Then we have a one-to-one correspondence between the two associated structures by the relation $g=L+\eta \otimes \eta ,$ where we denote by the same letter L the natural extension ($i_\xi L=0$) of the Levi form to a (0,2)-tensor field on M. So, we treat contact Riemannian structures together with strictly pseudo-convex almost CR structures. In earlier works [6,7,8, 10], the present author started the intriguing study of the interactions between them. For complex analytical considerations, it is desirable to have integrability of the almost complex structure J (on D). If this is the case, we speak of an (integrable) CR structure and of a CR manifold. Indeed, S. Webster [21, 22] introduced the term pseudo-Hermitian structure for a CR manifold with a non-degenerate Levi-form. In the present paper, we treat the pseudo-Hermitian structure as an extension to the case of non-integrable ${\mathscr {H}}$.

There is a canonical affine connection in a non-degenerate CR manifold, the so-called pseudo-Hermitian connection (or the Tanaka-Webster connection). S. Tanno [16] extends the Tanaka-Webster connection for strictly pseudo-convex almost CR manifolds (in which ${\mathscr {H}}$ is in general non-integrable). We call it the generalized Tanaka-Webster connection. Using this we have the pseudo-Hermitian Ricci curvature tensor. If the pseudo-Hermitian Ricci curvature tensor is a scalar (field) multiple of the Levi form in a strictly pseudo-convex almost CR manifold, then it is said to have the pseudo-Einstein structure. A pseudo-Hermitian CR space form is a strictly pseudo-convex CR manifold of constant holomorphic sectional curvature (for Tanaka-Webster connection). Then we have that a pseudo-Hermitian CR space form is pseudo-Einstein. In Sect. 4, we study the generalized Chern-Moser-Tanaka curvature tensor C as a pseudo-conformal invariant in a strictly pseudo-convex almost CR manifold. Then we first prove that the Chern-Moser-Tanaka curvature tensor vanishes for a pseudo-Hermitian CR space form. Moreover, we prove that for a strictly pseudo-convex almost CR manifold $M^{2n+1}$ ($n>1$) with vanishing C, M is pseudo-Einstein if and only if M is of pointwise constant holomorphic sectional curvature.

2 Preliminaries

We start by collecting some fundamental materials about contact Riemannian geometry and strictly pseudo-convex pseudo-Hermitian geometry. All manifolds in the present paper are assumed to be connected, oriented and of class $C^{\infty }$.

2.1 Contact Riemannian Structures

A contact manifold $(M,\eta )$ is a smooth manifold $M^{2n+1}$ equipped with a global one-form $\eta$ such that $\eta \wedge (d\eta )^n\ne 0$ everywhere on M. For a contact form $\eta$, there exists a unique vector field $\xi$, called the characteristic vector field, satisfying $\eta (\xi )=1$ and $d\eta (\xi ,X)=0$ for any vector field X. It is well-known that there also exist a Riemannian metric g and a (1, 1)-tensor field $\varphi$ such that

$$\begin{aligned} \eta (X)=g(X,\xi ),\ d\eta (X,Y)=g(X,\varphi Y), \ \varphi ^2 X=-X+\eta (X)\xi , \end{aligned}$$

(1)

where X and Y are vector fields on M. From (1), it follows that

$$\begin{aligned} \varphi \xi =0,\ \eta \circ \varphi =0, \ g(\varphi X,\varphi Y)=g(X,Y)-\eta (X)\eta (Y). \end{aligned}$$

(2)

A Riemannian manifold M equipped with structure tensors $(\eta ,g)$ satisfying (1) is said to be a contact Riemannian manifold or contact metric manifold and it is denoted by $M=(M;\eta ,g)$. Given a contact Riemannian manifold M, we define a (1, 1)-tensor field h by $h=\frac{1}{2} \pounds _{\xi }\varphi$, where $\pounds _{\xi }$ denotes Lie differentiation for the characteristic direction $\xi$. Then we may observe that h is self-adjoint and satisfies

$$\begin{aligned} h\xi =0,\quad h\varphi =-\varphi h, \end{aligned}$$

(3)

$$\begin{aligned} \nabla _X\xi =-\varphi X-\varphi hX, \end{aligned}$$

(4)

where $\nabla$ is Levi-Civita connection. From (3) and (4) we see that $\xi$ generates a geodesic flow. Furthermore, we know that $\nabla _\xi \varphi =0$ in general (cf. p. 67 in [1]). From the second equation of (3) it follows also that

$$\begin{aligned} (\nabla _\xi h)\varphi =-\varphi (\nabla _\xi h). \end{aligned}$$

(5)

A contact Riemannian manifold for which $\xi$ is Killing is called a K-contact manifold. It is easy to see that a contact Riemannian manifold is K-contact if and only if $h=0$. For further details on contact Riemannian geometry, we refer to [1].

2.2 Pseudo-Hermitian Almost CR Structures

For a contact manifold M, the tangent space $T_pM$ of M at each point $p\in M$ is decomposed as $T_pM=D_p\oplus \{\xi \}_p$ (direct sum), where we denote $D_p=\{v\in T_pM|\eta (v)=0\}$. Then the 2n-dimensional distribution (or subbundle) $D:p\rightarrow D_p$ is called the contact distribution (or contact subbundle). Its associated almost CR structure is given by the holomorphic subbundle

$${\mathscr {H}}=\{ X-i JX: X\in \varGamma (D) \}$$

of the complexification ${\mathbb C}TM$ of the tangent bundle TM, where $J=\varphi |D$, the restriction of $\varphi$ to D. Then we see that each fiber ${\mathscr {H}}_p$ $(p\in M)$ is of complex dimension n and ${\mathscr {H}}\cap \bar{\mathscr {H}}=\{0\}$. Furthermore, we have $\mathbb C D={\mathscr {H}}\oplus \bar{\mathscr {H}}$. For the real representation $\{D,J\}$ of ${\mathscr {H}}$ we define the Levi form by

$$ L: \varGamma (D)\times \varGamma (D)\rightarrow {\mathscr {F}}(M),\quad L(X,Y)=-d\eta (X,JY) $$

where ${\mathscr {F}}(M)$ denotes the algebra of differential functions on M. Then we see that the Levi form is Hermitian and positive definite. We call the pair $(\eta ,L)$ (or $(\eta ,J)$) a strictly pseudo-convex, pseudo-Hermitian structure on M. We say that the almost CR structure is integrable if $[{\mathscr {H}},{\mathscr {H}}]\subset {\mathscr {H}}$. Since $d\eta (JX,JY)=d\eta (X,Y)$, we see that $[JX,JY]-[X,Y]\in \varGamma (D)$ and $[JX,Y]+[X,JY]\in \varGamma (D)$ for $X,Y\in \varGamma (D)$, further if M satisfies the condition $[J,J](X,Y)=0$ for $X,Y\in \varGamma (D)$, then the pair $(\eta ,J)$ is called a strictly pseudo-convex (integrable) CR structure and $(M;\eta ,J)$ is called a strictly pseudo-convex CR manifold or a strictly pseudo-convex integrable pseudo-Hermitian manifold. A pseudo-Hermitian torsion is defined by $\tau =\varphi h$ (cf. [2]).

For a given strictly pseudo-convex pseudo-Hermitian manifold M, the almost CR structure is integrable if and only if M satisfies the integrability condition $\varOmega =0$, where $\varOmega$ is a (1,2)-tensor field on M defined by

$$\begin{aligned} \varOmega (X,Y)=(\nabla _{X}\varphi )Y-g(X+hX,Y)\xi +\eta (Y)(X+hX) \end{aligned}$$

(6)

for all vector fields X, Y on M (see [16], Proposition 2.1]). It is well known that for 3-dimensional contact Riemannian manifolds their associated CR structures are always integrable (cf. [16]).

A Sasakian manifold is a strictly pseudo-convex CR manifold whose characteristic flow is isometric (or equivalently, vanishing the pseudo-Hermitian torsion). From (6) it follows at once that a Sasakian manifold is also determined by the condition

$$\begin{aligned} (\nabla _X\varphi )Y=g(X,Y)\xi -\eta (Y)X \end{aligned}$$

(7)

for all vector fields X and Y on the manifold.

Now, we review the generalized Tanaka-Webster connection [16] on a strictly pseudo-convex almost CR manifold $M=(M;\eta ,J)$. The generalized Tanaka-Webster connection $\hat{\nabla }$ is defined by

$$ \hat{\nabla }_XY=\nabla _XY+ \eta (X)\varphi Y+(\nabla _X\eta )(Y)\xi -\eta (Y)\nabla _X\xi $$

for all vector fields X, Y on M. Together with (4), $\hat{\nabla }$ may be rewritten as

$$\begin{aligned} \hat{\nabla }_XY=\nabla _XY+B(X,Y), \end{aligned}$$

(8)

where we have put

$$\begin{aligned} B(X,Y)=\eta (X)\varphi Y+\eta (Y)(\varphi X+\varphi hX)-g(\varphi X+\varphi hX,Y)\xi . \end{aligned}$$

(9)

Then, we see that the generalized Tanaka-Webster connection $\hat{\nabla }$ has the torsion $\hat{T}(X,Y)=2g(X,\varphi Y)\xi +\eta (Y)\varphi hX-\eta (X)\varphi hY.$ In particular, for a K-contact manifold we get

$$\begin{aligned} B(X,Y)=\eta (X)\varphi Y+\eta (Y)\varphi X-g(\varphi X,Y)\xi . \end{aligned}$$

(10)

Furthermore, it was proved that

Proposition 1

([16]) The generalized Tanaka-Webster connection $\hat{\nabla }$ on a strictly pseudo-convex almost CR manifold $M=(M;\eta ,J)$ is the unique linear connection satisfying the following conditions:

(i) $\hat{\nabla }\eta =0$, $\hat{\nabla }\xi =0$;

(ii) $\hat{\nabla }g=0$, where g is the associated Riemannian metric;

$(iii-1)$ $\hat{T}(X,Y)=2L(X,JY)\xi$, $X,\ Y\in \varGamma (D)$;

$(iii-2)$ $\hat{T}(\xi ,\varphi Y)=-\varphi \hat{T}(\xi ,Y)$, $Y\in \varGamma (D)$;

(iv) $(\hat{\nabla }_X \varphi ) Y=\varOmega (X,Y)$, $X,\ Y\in \varGamma (TM)$.

The pseudo-Hermitian connection (or The Tanaka-Webster connection) [14, 22] on a non-degenerate (integrable) CR manifold is defined as the unique linear connection satisfying (i), (ii), (iii-1), (iii-2) and $\varOmega =0$. We refer to [2] for more details about pseudo-Hermitian geometry in strictly pseudo-convex almost CR manifolds.

2.3 Pseudo-homothetic Transformations

In this subsection, we first review

Definition 1

Let $(M;\eta ,\xi .\varphi ,g)$ be a contact Riemannian manifold. Then a diffeomorphism f on M is said to be a pseudo-homothetic transformation if there exists a positive constant a such that

$$\begin{aligned} f^{*}\eta =a\eta ,\ f_{*}\xi =\xi /a,\ \varphi \circ f_{*}=f_{*}\circ \varphi ,\ f^{*}g=ag+a(a-1)\eta \otimes \eta . \end{aligned}$$

Due to S. Tanno [15], we have

Theorem 1

If a diffeomorphism f on a contact Riemannian manifold M is $\varphi$-holomorphic, i.e.,

$$\begin{aligned} \varphi \circ f_{*}=f_{*}\circ \varphi , \end{aligned}$$

then f is a pseudo-homothetic transformation.

Here, the new contact Riemannian manifold $(M;\bar{\eta },\bar{\xi }.\bar{\varphi },\bar{g})$ defined by

$$\begin{aligned} \bar{\eta }=a\eta ,\ \bar{\xi }=\xi /a,\ \bar{\varphi }=\varphi ,\ \bar{g}=ag+a(a-1)\eta \otimes \eta , \end{aligned}$$

(11)

is called a pseudo-homothetic deformation of $(M,\eta ,\xi .\varphi ,g)$. Then we have

$$\begin{aligned} \bar{\nabla }_X Y=\nabla _X Y+A(X,Y), \end{aligned}$$

(12)

where A is the (1, 2)-type tensor defined by

$$A(X,Y)=-(a-1)[\eta (Y)\varphi X+\eta (X)\varphi Y]-\frac{a-1}{a}g(\varphi hX,Y)\xi .$$

Then we have

Proposition 2

([9]) The generalized Tanaka-Webster connection is pseudo-homothetically invariant.

The so-called $(k,\mu )$-spaces are defined by the condition

$$ R(X,Y)\xi =(kI+\mu h)(\eta (Y)X-\eta (X)Y)$$

for $(k,\mu )\in \mathbb {R}^2$, where I denotes the identity transformation. This class involves the Sasakian case for $k=1$ $(h=0)$. For a non-Sasakian contact Riemannian manifold, h has the only two eigenvalues $\sqrt{1-k}$ and $-\sqrt{1-k}$ on D with their multiplicities n respectively. The $(k,\mu )$-spaces have integrable CR structures and further, this class of spaces is invariant under pseudo-homothetic transformations. Indeed, a pseudo-homothetic transformation with constant $a(> 0)$ transforms a $(k,\mu )$-space into a $(\bar{k},\bar{\mu })$-space where $\bar{k}=\frac{k+a^2-1}{a^2} \quad \text { and } \quad \bar{\mu }=\frac{\mu +2a-2}{a}$ (cf. [1] or [3]). In particular, we find that $k=1$ and $\mu =2$ are the only two invariants under pseudo-homothetic transformations for all $a\ne 1$.

3 Pseudo-Einstein Structures

We define the pseudo-Hermitian curvature tensor (or the generalized Tanaka-Webster curvature tensor) on a strictly pseudo-convex almost CR manifold $\hat{R}$ of $\hat{\nabla }$ by

$$ \hat{R}(X,Y)Z=\hat{\nabla }_X(\hat{\nabla }_Y Z)-\hat{\nabla }_Y(\hat{\nabla }_X Z)-\hat{\nabla }_{[X,Y]}Z $$

for all vector fields X, Y, Z in M. We remark that the generalized Tanaka-Webster connection is not torsion-free, and then the Jacobi- or Bianchi-type identities do not hold, in general. From the definition of $\hat{R}$, we have

$$\begin{aligned} \hat{R}(X,Y)Z= R(X,Y)Z+H(X,Y)Z, \end{aligned}$$

(13)

and

$$\begin{aligned} H(X,Y)Z =&\ \eta (Y)\big ((\nabla _X\varphi )Z-g(X+hX,Z)\xi \big )-\eta (X)\big ((\nabla _Y\varphi )Z-g(Y+hY,Z)\xi \big )\nonumber \\&+\eta (Z)\big ((\nabla _X\varphi )Y-(\nabla _Y\varphi )X+(\nabla _X\varphi h)Y-(\nabla _Y\varphi h)X\nonumber \\&+\eta (Y)(X+hX)-\eta (X)(Y+hY)\big )-2g(\varphi X,Y)\varphi Z\\ {}&-g(\varphi X+\varphi hX,Z)(\varphi Y+\varphi hY)+g(\varphi Y+\varphi hY,Z)(\varphi X+\varphi hX)\nonumber \\&-g((\nabla _X\varphi )Y-(\nabla _Y\varphi )X+(\nabla _X\varphi h)Y-(\nabla _Y\varphi h)X,Z)\xi \nonumber \end{aligned}$$

(14)

for all vector fields X, Y, Z in M.

Now, we introduce the pseudo-Hermitian Ricci (curvature) tensor:

$$ \hat{\rho }(X,Y)=\frac{1}{2}\hbox {trace of }\{V\mapsto J\hat{R}(X,JY)V\}, $$

where X, Y are vector fields orthogonal to $\xi$. This definition was referred as a 2nd kind in the author’s earlier work [9]. Indeed, the pseudo-Hermitian Ricci (curvature) tensor of the 1st kind $\hat{\rho }_1$ is defined by

$$ \hat{\rho }_1(X,Y)=\hbox {trace of }\{V\mapsto \hat{R}(V,X)Y\}, $$

where V is any vector field on M and X, Y are vector fields orthogonal to $\xi$. Then we can find the following useful relation between the two notions in general:

$$\begin{aligned} \begin{aligned} \hat{\rho }(X,Y)=&\hat{\rho }_1(X,Y)-2(n-1)g(hX,Y)\\ {}&\quad +\sum _{i=1}^{2n}\Big (g((\hat{\nabla }_{e_i}\varOmega )(X,Y),\varphi e_i)-g((\hat{\nabla }_{X}\varOmega )(e_i,Y),\varphi e_i)\Big ) \end{aligned} \end{aligned}$$

(15)

for $X,Y \in \varGamma (D)$ (cf. [17]). We define the corresponding pseudo-Hermitian Ricci operator $\hat{Q}$ is defined by $L(\hat{Q}X,Y)= \hat{\rho }(X,Y)$. The Tanaka-Webster (or the pseudo-Hermitian) scalar curvature $\hat{r}$ is given by

$$ \hat{r}=\hbox {trace of }\{V\mapsto \hat{Q} V\}. $$

Then, from Proposition 2, we get

Corollary 1

The pseudo-Hermitian curvature tensor (or The generalized Tanaka-Webster curvature tensor) $\hat{R}$ and the pseudo-Hermitian Ricci tensor $\hat{Q}$ are pseudo-homothetic invariants.

Definition 2

Let $(M;\eta ,J)$ be a strictly pseudo-convex almost CR manifold. Then the pseudo-Hermitian structure $(\eta ,J)$ is said to be pseudo-Einstein if the pseudo-Hermitian Ricci tensor is proportional to the Levi form, namely,

$$ \hat{\rho }(X,Y)=\lambda L(X,Y), $$

where $X,Y\in \varGamma (D)$, where $\lambda =\hat{r}/2n$.

Remark 1

N. Tanaka [13] and J.M. Lee [11] defined the pseudo-Hermitian Ricci tensor on a non-degenerate CR manifold in a complex fashion. Further, J.M. Lee defined and intensively studied the pseudo-Einstein structure. Then every 3-dimensional strictly pseudo-convex CR manifold is pseudo-Einstein.

Remark 2

From (15), we at once see that for the Sasakian case or the 3-dimensional case $\hat{\rho }=\hat{\rho }_1$.

Moreover, we have

Proposition 3

([9]) A non-Sasakian contact $(k,\mu )$-space $(k<1)$ is pseudo-Einstein with constant pseudo-Hermitian scalar curvature $\hat{r}=2n^2(2-\mu )$.

In [3] they proved that unit tangent sphere bundles with standard contact metric structures are $(k,\mu )$-spaces if and only if the base manifold is of constant curvature b with $k=b(2-b)$ and $\mu =-2b$. Thus, we have

Corollary 2

The standard contact metric structure of $T_1 M(b)$ of a space of constant curvature b is pseudo-Einstein. Its pseudo-Hermitian scalar curvature $\hat{r}=4n^2(1+b)$.

The class of contact $(k,\mu )$-spaces, whose associated CR structures are integrable as stated at the end of Sect. 2, contains non-unimodular Lie groups with left-invariant contact metric structure other than unit tangent bundles of a space of constant curvature (see [4]).

4 Pseudo-Hermitian CR Space Forms

In this section, we give

Definition 3

([7]) Let $(M;\eta ,J)$ be a strictly pseudo-convex almost CR manifold. Then M is said to be of constant holomorphic sectional curvature c (with respect to the generalized Tanaka-Webster connection) if M satisfies

$$L(\hat{R}(X,\varphi X)\varphi X,X)=c$$

for any unit vector field X orthogonal to $\xi$. In particular, for the CR integrable case we call M a pseudo-Hermitian (strictly pseudo-convex) CR space form.

Then for a strictly pseudo-convex almost CR manifold M, from (13) and (14) we get

$$\begin{aligned} g(\hat{R}(X,\varphi X)\varphi X,X)=g(R(X,\varphi X)\varphi X,X)+3g(X,X)^2-g(hX,X)^2-g(\varphi hX,X)^2 \end{aligned}$$

(16)

for any X orthogonal to $\xi$. From this, we easily see that s Sasakian space form $M^{2n+1}(c_0)$ of constant $\varphi$-holomorphic sectional curvature $c_0$ (with respect to the Levi-Civita connection) is a strictly pseudo-convex CR space form of constant holomorphic sectional curvature (with respect to the Tanaka-Webster connection) $c=c_0+3$. Simply connected and complete Sasakian space forms are the unit sphere $S^{2n+1}$ with the natural Sasakian structure with $c_0=1$ $(c=4)$, the Heisenberg group $H^{2n+1}$ with Sasakian $\varphi$-holomorphic sectional curvature $c_0=-3$ $(c=0)$, or $B^n\times R$ with Sasakian $\varphi$-holomorphic sectional curvature $c_0=-7$ $(c=-4)$, where $B^n$ is a simply connected bounded domain in $C^n$ with constant holomorphic sectional curvature $-4$.

For a class of the contact $(k,\mu )$-spaces, we proved the following results.

Theorem 2

([7]) Let M be a contact $(k,\mu )$-space. Then M is of constant holomorphic sectional curvature c for Tanaka-Webster connection if and only if (1) M is Sasakian space of constant $\varphi$-holomorphic sectional curvature $c_0=c-3$, (2) $\mu =2$ and $c=0$, or (3) dim M=3 and $\mu =2-c$.

Corollary 3

([7]) The standard strictly pseudo-convex CR structure on a unit tangent sphere bundle $T_1 M(b)$ of $(n+1)$-dimensional space of constant curvature b has constant holomorphic sectional curvature c if and only if $b=-1$ and $c=0$, or $n=1$ and $b=(c-2)/2$.

Remark 3

(1) The standard contact metric structure of the unit tangent sphere bundle $T_1 \mathbb {S}^{n+1}(1)$ is Sasakian [20], but it has not constant holomorphic sectional curvature for both Levi-Civita and Tanaka-Webster connection.

(2) The unit tangent sphere bundle $T_1 \mathbb {H}^{n+1}(-1)$ of a hyperbolic space $\mathbb {H}^{n+1}(-1)$ is a non-Sasakian example of constant holomorphic sectional curvature for Tanaka-Webster connection but not for Levi-Civita connection.

In [7] we determined the Riemannian curvature tensor explicitly for a strictly pseudo-convex CR space of constant holomorphic sectional curvature c. Then we have

$$\begin{aligned} \begin{aligned} g(&\hat{R}(X,Y)Z,W)=g(H(X,Y)Z,W)+\frac{c}{4}\Big \{g(Y,Z)g(X,W) -g(X,Z)g(Y,W)\\&+g(\varphi Y,Z)g(\varphi X,W)-g(\varphi X,Z)g(\varphi Y,W)-2g(\varphi X,Y)g(\varphi Z,W)\Big \}\\ \end{aligned} \end{aligned}$$

(17)

for all vector fields $X,Y,Z,W \perp \xi$, where

$$\begin{aligned} \begin{aligned} g(H(X,Y)Z,W) =&\,g(Y,Z)g(hX,W)-g(X,Z)g(hY,W)\\&-g(Y,W)g(hX,Z)+g(X,W)g(hY,Z)\\&+g(\varphi Y,Z)g(\varphi hX,W)-g(\varphi X,Z)g(\varphi hY,W)\\&-g(\varphi Y,W)g(\varphi hX,Z)+g(\varphi X,W)g(\varphi hY,Z). \end{aligned} \end{aligned}$$

(18)

Then from (17) we get

$$\begin{aligned} \hat{\rho }(X,Y)=c(n+1)/2\ g(X,Y). \end{aligned}$$

(19)

Proposition 4

([9]) A strictly pseudo-convex CR space form of constant holomorphic sectional curvature c is pseudo-Einstein with constant pseudo-Hermitian scalar curvature $\hat{r}=n(n+1)c$.

5 The Chern-Moser-Tanaka Invariant

Now, we review the pseudo-conformal transformations of a strictly pseudo-convex almost CR structure. Given a contact form $\eta$, we consider a 1-form $\bar{\eta }=\sigma \eta$ for a positive smooth function $\sigma$. By assuming $\bar{\phi }|D=\phi |D$ ($\bar{J}=J$), the associated Riemannian structure $\bar{g}$ of $\bar{\eta }$ is determined in a natural way. Namely, we have

$$ \begin{aligned} \bar{\xi }=(1/\sigma )&(\xi +\zeta ),\ \zeta =(1/2\sigma )\phi (\text {grad}\ \sigma ),\ \bar{\phi }=\phi +(1/2\sigma )\eta \otimes (\text {grad}\ \sigma -\xi \sigma \cdot \xi ),\\&\quad \bar{g}{=}\sigma g-\sigma (\eta \otimes \nu +\nu \otimes \eta )+\sigma (\sigma -1+\Vert \zeta \Vert ^2)\eta \otimes \eta , \end{aligned} $$

where $\nu$ is dual to $\zeta$ with respect to g. We call the transformation $(\eta ,J)\rightarrow (\bar{\eta },\bar{J})$ a pseudo-conformal transformation (or gauge transformation) of the strictly pseudo-convex almost CR structure. We remark in particular that when $\sigma$ is a constant, then a gauge transformation reduces to a pseudo-homothetic transformation.

Let $\omega$ be a nowhere vanishing $(2n+1)$-form on M and fix it. Let $dM(g)=((-1)^n/2^n n!)\eta \wedge (d\eta )^n$ denote the volume element of $(M,\eta ,g)$. We define $\beta$ by $dM(g)=\pm e^\beta \omega$ and $\theta \in \varGamma (D^*)$ by $\theta (X)=X\beta$ for $X\in \varGamma (D)$. For a strictly pseudo-convex almost CR manifold, the generalized Chern-Moser-Tanaka curvature tensor $C\in \varGamma (D\otimes {D^*}^3)$ is defined by S. Tanno in [18] (see also, [8]).

$$\begin{aligned} \begin{aligned} (2n+4)&g(C(X,Y)Z,W)\\ =&\,(2n+4)g(\hat{R}(X,Y)Z,W)\\&-\hat{\rho }(Y,Z)g(X,W)+\hat{\rho }(X,Z)g(Y,W)-g(Y,Z)\hat{\rho }(X,W)+g(X,Z)\hat{\rho }(Y,W)\\&+\hat{\rho }(Y,\varphi Z)g(\varphi X,W)-\hat{\rho }(X,\varphi Z)g(\varphi Y,W)-[\hat{\rho }(X,\varphi Y)-\hat{\rho }(\varphi X,Y)]g(\varphi Z,W)\\&+\hat{\rho }(X,\varphi W)g(\varphi Y,Z)-\hat{\rho }(Y,\varphi W)g(\varphi X,Z)-[\hat{\rho }(Z,\varphi W)-\hat{\rho }(\varphi Z,W)]g(\varphi X,Y)\\&+[\hat{r}/(2n+2)][g(Y,Z)g(X,W)-g(X,Z)g(Y,W)\\&+g(\varphi Y,Z)g(\varphi X,W)-g(\varphi X,Z)g(\varphi Y,W) -2g(\varphi X,Y)g(\varphi Z,W)] \\&-(2n+4)[g(hY,Z)g(X,W)-g(hX,Z)g(Y,W)+g(Y,Z)g(hX,W)\\&-g(X,Z)g(hY,W)+g(\varphi hY,Z)g(\varphi X,W)-g(\varphi hX,Z)g(\varphi Y,W)\\&+g(\varphi hX,W)g(\varphi Y,Z)-g(\varphi hY,W)g(\varphi X,Z)]\\&-(n+2)/(n+1)g(U(X,Y,Z;\theta ),W)). \end{aligned} \end{aligned}$$

(20)

Here $U\in \varGamma (D^2\otimes {D^*}^3)$ and $U(X,Y,Z;\theta )=(\theta _j U^{ji}_{lhk}X^h Y^k Z^l)$ in terms of an adapted frame $\{e_\alpha \}=\{e_j,\ e_0=\xi ;\ 1\le j\le 2n\}$. For a full understanding, we may describe it by using the components of U in terms of $\{e_j,e_0\}$ (cf. [18]). That is,

$$\begin{aligned} \begin{aligned} U^{ji}_{lhk}=&\,2\Big [1/(n+2)\{-{\delta }^i_h(\varOmega ^j_{km}+\varOmega ^j_{mk}){\phi }^m_l-{\phi }^i_h(\varOmega ^j_{lk}+\varOmega ^j_{kl}) +g_{hl}(\varOmega ^j_{km}+\varOmega ^j_{mk}){\phi }^{mi}\\&-{\phi }_{hl}(\varOmega ^j_{km}+\varOmega ^j_{mk})g^{mi}\} +\varOmega ^j_{lk}{\phi }^i_h+{\phi }_{hl}\varOmega ^j_{mk}g^{im}+\varOmega ^j_{hk}{\phi }^i_l\\&+(1/2)(\varOmega ^j_{ml}-\varOmega ^j_{lm})g^{mi}{\phi }_{hk}+{\phi }^j_l \varOmega ^i_{hk}+{\phi }^j_h \varOmega ^i_{lk}-(1/2){\phi }^{ij}\varOmega ^m_{kl}g_{hm}\Big ]_{hk}, \end{aligned} \end{aligned}$$

where $[\cdots ]_{hk}$ denotes the skew-symmetric part of $[\cdots ]$ with respect to h, k.

Remark 4

(1) If $n=1$ (dim M=3), then we always have $C=0$ (see Remark in [18]).

(2) When $(M;\eta ,g)$ is Sasakian, then ($h=0$ and) C reduces to the C-Bochner curvature tensor, which is the corresponding (through the Boothby-Wang fibration) to the Bochner curvature tensor in a Kähler manifold [12].

Using (17) and (19), from the Eq. (9) we find

Proposition 5

On a pseudo-Hermitian CR space form, the Chern-Moser-Tanaka invariant C vanishes.

Moreover we have

Theorem 3

Let $(M^{2n+1};\eta ,J)$ $(n>1)$ be a strictly pseudo-convex almost CR manifold with vanishing C. Then M is pseudo-Einstein if and only if M is of pointwise constant holomorphic sectional curvature for the Tanaka-Webster connection.

The argument and computation of present paper gives a simpler proof of [9, Theorem 22].

Remark 5

The unit tangent sphere bundle $T_1 \mathbb {H}^{n+1}(-1)$ of a hyperbolic space $\mathbb {H}^{n+1}(-1)$ is a non-Sasakian example which supports Theorem 3 well. It was proved that the Chern-Moser-Tanaka curvature tensor C on $T_1 \mathbb {H}^{n+1}(-1)$ vanishes [19] and within the class of $(k,\mu )$-spaces, it is the only such an example [8].