On the geometry of trans-para-Sasakian manifolds

TL;DR

This paper introduces trans-para-Sasakian manifolds, exploring their geometric properties and curvature, and establishing conditions for them to be η-Einstein or Einstein, expanding the understanding of these structures.

Contribution

It defines trans-para-Sasakian manifolds and investigates their curvature properties, providing new conditions for Einstein and η-Einstein classifications.

Findings

Identification of curvature properties of trans-para-Sasakian manifolds

Conditions for manifolds to be η-Einstein or Einstein

Extension of trans-Sasakian geometry to the para setting

Abstract

In this paper, we introduce the trans-para-Sasakian manifolds and we study their geometry. These manifolds are an analogue of the trans-Sasakian manifolds in the Riemannian geometry. We shall investigate many curvature properties of these manifolds and we shall give many conditions under which the manifolds are either $\eta-$Einstein or Einstein manifolds.

Equations

$$\displaystyle\begin{array}[]{cl}(i)&\varphi(\xi)=0,\quad\eta\circ\varphi=0,\\[5.0pt] (ii)&\eta(\xi)=1\quad\varphi^{2}=id-\eta\otimes\xi,\\[5.0pt] (iii)&\textrm{distribution $\mathbb{D}:p\in M\longrightarrow\mathbb{D}_{p}\subset T_{p}M:$}\\[1.0pt] &\textrm{$\mathbb{D}_{p}=Ker\eta=\{X\in T_{p}M:\eta(X)=0\}$ is called {\it paracontact}}\\[1.0pt] &\textrm{{\it distribution} generated by $\eta$.}\end{array}$$

$$g(\phi X,\phi Y)=-g(X,Y)+\eta (X)\eta (Y),$$

$$({\nabla }_X\phi )Y=-g(X,Y)\xi +\eta (Y)X,$$

$${\nabla }_X\xi =-\alpha \phi X-\beta (X-\eta (X)\xi ).$$

$$({\nabla }_X\eta )Y=\alpha g(X,\phi Y)-\beta (g(X,Y)-\eta (X)\eta (Y)),$$

$$d\eta (X,Y)=\alpha g(X,\phi Y),$$

$$({\pounds }_{\xi }g)(X,Y)=-2\beta (g(X,Y)-\eta (X)\eta (Y)),$$

$${\pounds }_{\xi }\phi =0,$$

$${\pounds }_{\xi }\eta =0,$$

$$E_1=e^z(\frac{\partial }{\partial x}+y\frac{\partial }{\partial z}).E_2=e^z\frac{\partial }{\partial y},E_3=\frac{\partial }{\partial z},$$

$$\begin{array}[]{llll}\varphi E_{1}=E_{2},\quad\varphi E_{2}=E_{1},\quad\varphi E_{3}=0\\ \xi=E_{3},\quad\eta(E_{3})=1,\quad\eta(E_{1})=\eta(E_{2})=0,\\ g(E_{1},E_{1})=g(E_{3},E_{3})=-g(E_{2},E_{2})=1,\\ \quad g(E_{i},E_{j})=0,\quad i\neq j\in\{1,2,3\}.\end{array}$$

$$\begin{array}[]{llll}[E_{1},E_{2}]=ye^{z}E_{2}-e^{2z}E_{3},\quad[E_{2},E_{3}]=-E_{2},\quad[E_{1},E_{3}]=-E_{3}.\end{array}$$

$$\begin{array}[]{llll}\nabla_{E_{1}}E_{1}=E_{3},\quad\nabla_{E_{1}}E_{2}=-\frac{1}{2}e^{2z}E_{3},\quad\nabla_{E_{1}}E_{3}=-E_{1}-\frac{1}{2}e^{2z}E_{2},\\ \nabla_{E_{2}}E_{1}=-ye^{z}E_{2}+\frac{1}{2}e^{2z}E_{3},\quad\nabla_{E_{2}}E_{2}=-ye^{z}E_{1}-E_{3},\quad\nabla_{E_{2}}E_{3}=-\frac{1}{2}e^{2z}E_{1}-E_{2},\\ \nabla_{E_{3}}E_{1}=-\frac{1}{2}e^{2z}E_{2},\quad\nabla_{E_{3}}E_{2}=-\frac{1}{2}e^{2z}E_{1},\quad\nabla_{E_{3}}E_{3}=0.\\ \end{array}$$

$$({\nabla }_{E_1}\eta )E_1=-\beta =-1,({\nabla }_{E_2}\eta )E_1=-\alpha =-\frac{1}{2}{e}^{2z},({\nabla }_{E_3}\eta )E_1=0.$$

$$R(X,Y)\xi =-({\alpha }^2+{\beta }^2)(\eta (Y)X-\eta (X)Y)-2\alpha \beta (\eta (Y)\phi X-\eta (X)\phi Y)-$$

$$-X(\alpha )\phi Y+Y(\alpha )\phi X+Y(\beta ){\phi }^{2}X-X(\beta ){\phi }^{2}Y.$$

$${\nabla }_X{\nabla }_Y\xi ={\nabla }_X(-\alpha \phi Y-\beta (Y-\eta (Y)\xi )=$$

$$=-X(\alpha )\phi Y-\alpha {\nabla }_X\phi Y-X(\beta ){\phi }^{2}Y-\beta {\nabla }_{X}Y-\beta (X\eta (Y))\xi -$$

$$-\alpha \beta \eta (Y)\phi X-{\beta }^2\eta (Y)X+{\beta }^2\eta (X)\eta (Y)\xi ,$$

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

$$=-X(\alpha )\phi Y+Y(\alpha )\phi X-\alpha (({\nabla }_X\phi )Y-({\nabla }_Y\phi )X)-$$

$$-X(\beta ){\phi }^{2}Y+Y(\beta ){\phi }^{2}X+\beta (({\nabla }_X\eta )Y-({\nabla }_Y\eta )X)\xi -$$

$$-\alpha \beta (\eta (Y)\phi X-\eta (X)\phi Y)-{\beta }^2(\eta (Y)X-\eta (X)Y),$$

$$R(\xi ,X)\xi =({\alpha }^2+{\beta }^2-\xi (\beta ))(X-\eta (X)\xi ),$$

$$2\alpha \beta -\xi (\alpha )=0.$$

$$R(\xi ,Z)X=-({\alpha }^2+{\beta }^2)(g(X,Z)-\eta (X)Z)-2\alpha \beta (g(\phi X,Z)\xi +\eta (X)\phi Z)+$$

$$+X(\alpha )\phi Z+g(\phi X,Z)grad\alpha -X(\beta )(Z-\eta (Z)\xi )-g(\phi X,\phi Z)grad\beta .$$

$$R(\xi ,X)\xi =({\alpha }^2+{\beta }^2-\xi (\beta ))(X-\eta (X)\xi )+(2\alpha \beta -\xi (\alpha ))\phi Y,$$

$$R(\xi ,X)\xi =({\alpha }^2+{\beta }^2-\xi (\beta ))(X-\eta (X)\xi )-(2\alpha \beta -\xi (\alpha ))\phi Y.$$

$$Ric(X,\xi )=-(2n({\alpha }^2+{\beta }^2)-\xi (\beta ))\eta (X)+(2n-1)X(\beta )-\phi X(\alpha ),$$

Full text

1. Introduction

In Grey-Hervella classification of almost Hermitian manifolds (see [3]), there appears a class, $\mathcal{W}_{4}$, of Hermitian manifolds which are closely related to locally conformal $K\ddot{a}hler$ manifolds. An almost contact structure on a manifold $M$ is called a trans-Sasakian structure (see [8]) if the product manifold $M\times\mathbb{R}$ belongs to the class $\mathcal{W}_{4}$. The class $\mathcal{C}_{6}\bigoplus\mathcal{C}_{5}$ (see [6], [7]) coincides with the class of trans-Sasakian structures of type $(\alpha,\beta)$. In fact, in (see [7]), local nature of the two subclasses, namely the $\mathcal{C}_{5}$ and the $\mathcal{C}_{6}$ structures, of trans-Sasakian structures are characterized completely. We note the that trans-Sasakian structures of type $(0,0)$, $(0,\beta)$ and $(\alpha,0)$ are cosympletic (see [1]), $\beta-$Kenmotsu (see [4]) and $\alpha-$Sasakian (see [4]), respectively. We consider the trans-para-Sasakian manifolds as an analogue of the trans-Sasakian manifolds. A trans-para-Sasakian manifold is a trans-para-Sasakian structure of type $(\alpha,\beta)$, where $\alpha$ and $\beta$ are smooth functions. The trans-para-Sasakian manifolds of types $(\alpha,\beta)$, and are respecively the para-cosympletic, para-Sasakian (in case $\alpha=1$, these are just the para-Sasakian manifolds; in case $\alpha=-1$, these are the quasi-para-Sasakian manifolds, see [11]) and para-Kenmotsu (for the case $\beta=1$ see [12]). In the second section, we give the formal definition of trans-para-Sasakian manifolds of type $(\alpha,\beta)$ and we prove some basic properties. We give an example for a 3-dimensional trans-para-Sasakian manifold. In the last section, we investigate the curvature properties of the trans-para-Sasakian manifolds. Further, we find many conditions under which the manifolds are either $\eta-$Einstein or Einstein manifolds.

2. Preliminaries

A (2n+1)-dimensional smooth manifold $M^{(2n+1)}$ has an almost paracontact structure $(\varphi,\xi,\eta)$ if it admits a tensor field $\varphi$ of type $(1,1)$, a vector field $\xi$ and a 1-form $\eta$ satisfying the following compatibility conditions

[TABLE]

The tensor field $\varphi$ induces an almost paracomplex structure [5] on each fibre on $\mathbb{D}$ and $(\mathbb{D},\varphi,g_{|\mathbb{D}})$ is a $2n$-dimensional almost paracomplex distribution. Since $g$ is non-degenerate metric on $M$ and $\xi$ is non-isotropic, the paracontact distribution $\mathbb{D}$ is non-degenerate.

An immediate consequence of the definition of the almost paracontact structure is that the endomorphism $\varphi$ has rank $2n$, $\varphi\xi=0$ and $\eta\circ\varphi=0$, (see [1, 2] for the almost contact case).

If a manifold $M^{(2n+1)}$ with $(\varphi,\xi,\eta)$-structure admits a pseudo-Riemannian metric $g$ such that

[TABLE]

then we say that $M^{(2n+1)}$ has an almost paracontact metric structure and $g$ is called compatible. Any compatible metric $g$ with a given almost paracontact structure is necessarily of signature $(n+1,n)$.

Note that setting $Y=\xi$, we have $\eta(X)=g(X,\xi).$

Further, any almost paracontact structure admits a compatible metric.

Definition 2.1**.**

If $g(X,\varphi Y)=d\eta(X,Y)$ (where $d\eta(X,Y)=\frac{1}{2}(X\eta(Y)-Y\eta(X)-\eta([X,Y])$ then $\eta$ is a paracontact form and the almost paracontact metric manifold $(M,\varphi,\eta,\xi,g)$ is said to be a paracontact metric manifold.

A paracontact metric manifold for which $\xi$ is Killing is called a $K-\emph{paracontact}$ manifold. A paracontact structure on $M^{(2n+1)}$ naturally gives rise to an almost paracomplex structure on the product $M^{(2n+1)}\times\Re$. If this almost paracomplex structure is integrable, then the given paracontact metric manifold is said to be a para-Sasakian. Equivalently, (see [10]) a paracontact metric manifold is a para-Sasakian if and only if

[TABLE]

for all vector fields $X$ and $Y$ (where $\nabla$ is the Livi-Civita connection of $g$).

Definition 2.2**.**

If $(\nabla_{X}\varphi)Y=\alpha(-g(X,Y)\xi+\eta(Y)X)+\beta(g(X,\varphi Y)\xi+\eta(Y)\varphi X),$ then the manifold $(M^{(2n+1)},\varphi,\eta,\xi,g)$ is said to be a trans-para-Sasakian manifold.

From $Definition\leavevmode\nobreak\ \ref{d1}$ we have

[TABLE]

Definition 2.3**.**

A $(2n+1)$-dimensional almost paracontact metric manifold is called

normal if $N(X,Y)-2d\eta(X,Y)\xi=0$, where $N(X,Y)=\varphi^{2}[X,Y]+[\varphi X,\varphi Y]-\varphi[\varphi X,Y]-\varphi[X,\varphi Y]$ is the Nijenhuis torsion tensor of $\varphi$ (see [10]).

Denoting by $\pounds$ the Lie differentiation of $g$, we see

Proposition 2.4**.**

Let $(M^{(2n+1)},\varphi,\eta,\xi,g)$ be a trans-para-Sasakian manifold. Then we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $X,Y\in T_{p}M.$

Since the proof of $Proposition\leavevmode\nobreak\ \ref{pro1}$ follows by routine calculation, we shall omit it.

From $Proposition\leavevmode\nobreak\ \ref{pro1}$ we see that $(M^{(2n+1)},\varphi,\eta,\xi,g)$ is normal.

Example 2.5*.*

Let us consider the 3-dimensional manifold $M^{3}=\{(x,y,z):(x,y,z)\in\Re^{3}_{1}\},z\neq 0$, where $(x,y,z)$ are the standard coordinates in $\Re^{3}_{1}$.We choose the vector fields

[TABLE]

which are linearly independent at each point of $M$. We define an almost paracontact structure $(\varphi,\xi,\eta)$ and a pseudo-Riemannian metric $g$ in the following way:

[TABLE]

By the definition of Lie bracket, we have

[TABLE]

Then $(M,\varphi,\xi,\eta,g)$ is a 3-dimensional almost paracontact manifold. The Koszul equality becomes

[TABLE]

We have $\nabla_{E_{1}}\xi=-\alpha\varphi E_{1}-\beta E_{2},\quad\nabla_{E_{2}}\xi=-\alpha\varphi E_{2}-\beta E_{2},\quad\nabla_{\xi}\xi=0$ for $E_{3}=\xi$, where $\alpha=\frac{1}{2}e^{2z}$ and $\beta=1$.

Again, by virtue of (2.10) and $(\nabla_{X}\eta)Y=X(\eta(Y))-\eta(\nabla_{X}Y)$ we obtain

[TABLE]

Thus from above the calculation the condition (2.9) and (2.10) are satisfied and the structure $(\varphi,\xi,\eta,g)$ is a trans-para-Sasakian structure of type $(\alpha,\beta)$, where $\alpha=\frac{1}{2}e^{2z}$ and $\beta=1$. Consequently $(M^{3},\varphi,\xi,\eta,g)$ is a trans-para-Sasakian manifold.

Finally, the sectional curvature $K(\xi,X)=\epsilon_{X}R(X,\xi,\xi,X)$, where $|X|=\epsilon_{X}=\pm 1$, of a plane section spanned by $\xi$ and the vector $X$ orthogonal to $\xi$ is called $\xi$-sectional curvature, where denoting by $R$ the curvature tensor of $\nabla$.

3. Some curvatureb properties of trans-para-Sasakian manifolds

We begin with the following Lemma.

Lemma 3.1**.**

Let $(M^{(2n+1)},\varphi,\eta,\xi,g)$ be a trans-para-Sasakian manifold. Then we have

[TABLE]

[TABLE]

Proof.

Using $Definition\leavevmode\nobreak\ \ref{d1}$, we obtain

[TABLE]

[TABLE]

[TABLE]

From here and (2.9), we get

[TABLE]

[TABLE]

[TABLE]

[TABLE]

which in view of $Definition\leavevmode\nobreak\ \ref{d1}$ and (2.10) gives (3.15). ∎

$Lemma\leavevmode\nobreak\ \ref{lem1}$ yields the following

Proposition 3.2**.**

If $(M^{(2n+1)},\varphi,\eta,\xi,g)$ is a trans-para-Sasakian manifold, then it is of $\xi-$sectional curvature $K(\xi,X)=-\epsilon_{X}(\alpha^{2}+\beta^{2}-\xi(\beta))$.

In a trans-para-Sasakian manifolds the functions $\alpha$ and $\beta$ can not be arbitrary. This fact is shown in the following

Theorem 3.3**.**

In trans-para-Sasakian manifold, we have

[TABLE]

[TABLE]

Proof.

Using (3.15) in $R(\xi,Z,X,Y)=R(X,Y,\xi,Z)$, we get

[TABLE]

[TABLE]

From (3.15), we get

[TABLE]

while gives us (3.15)

[TABLE]

The above two equations provide (3.16) and (3.17). ∎

From $Lemma\leavevmode\nobreak\ \ref{lem1}$, we have the following

Proposition 3.4**.**

In a $(2n+1)-$dimensional tras-para-Sasakian manifold, we have

[TABLE]

[TABLE]

where $Ric$ is the Ricci tensor and $Q$ is the Ricci operator given by

[TABLE]

Corollary 3.5**.**

If in a $(2n+1)-$dimensional trans-para-Sasakian manifold we have $\varphi(grad\alpha)=-(2n-1)grad\beta$, then

[TABLE]

and hence

[TABLE]

[TABLE]

From here on, we shall assume that $\varphi(grad\alpha)=-(2n-1)grad\beta$.

The Weyl-projective curvature tensor $P$ is defined as

[TABLE]

Hence we can state the following

Theorem 3.6**.**

A Weyl projectively flat trans-para-Sasakian manifold is an Einstein manifold.

Proof.

Suppose that $P=0$. Then from equation (3.24), we have

[TABLE]

From (3.25), we obtain

[TABLE]

Putting $W=\xi$ in (3.26), we get

[TABLE]

Again taking $X=\xi$, and using (3.15) and (3.22), we get

[TABLE]

∎

Theorem 3.7**.**

A trans-para-Sasakian manifold satisfying $R(X,Y)P=0$ is an Einstein manifold and also it is a manifold of scalar curvature $scal=-2n(2n+1)(\alpha^{2}+\beta^{2})$.

Proof.

Using (3.15) and (3.22) in (3.24), we get

[TABLE]

and

[TABLE]

Now,

[TABLE]

[TABLE]

By assumption $R(X,Y)P=0$, so we have

[TABLE]

[TABLE]

Therefore

[TABLE]

[TABLE]

From this, it follows that,

[TABLE]

[TABLE]

[TABLE]

Let $\{e_{i}\}$, $i=1,...,2n+1$ be an orthonormal basis. Then summing up for $1\leq i\leq 2n+1$ of the relation (3.32) for $Y=U=e_{i}$ yields

[TABLE]

From (3.30), we have

[TABLE]

Taking $Z=\xi$ in (3.34) and using (3.22) we obtain

[TABLE]

∎

The Weyl-conformal tensor $C$ is defined by

[TABLE]

[TABLE]

We have the following

Theorem 3.8**.**

A conformally flat trans-para-Sasakian manifold is an $\eta-$Einstein manifold.

Proof.

Suppose that $C=0$. Then from (3.36), we get

[TABLE]

[TABLE]

From the identity (3.37), we have

[TABLE]

[TABLE]

Again taking $X=\xi$ in (3.38), and using (3.15) and (3.22) we get

[TABLE]

∎

Theorem 3.9**.**

A trans-para-Sasakian manifold satisfying $R(X,Y)C=0$ is an $\eta-$Einstein manifold.

Proof.

From identity (3.36), we have $\eta(C(X,Y)\xi)=0$ and

[TABLE]

[TABLE]

Now,

[TABLE]

[TABLE]

By assumption $R(X,Y)C=0$, so we have

[TABLE]

[TABLE]

Therefore

[TABLE]

[TABLE]

From this, it follows that,

[TABLE]

[TABLE]

[TABLE]

Let $\{e_{i}\}$, $i=1,...,2n+1$ be an orthonormal basis. Then summing up for $1\leq i\leq 2n+1$ of the relation (3.42) for $Y=U=e_{i}$ yields

[TABLE]

From (3.40), we have

[TABLE]

∎

The concicular curvature tensor $\overline{C}$ is defined by

[TABLE]

We have the following

Theorem 3.10**.**

A trans-para-Sasakian manifold satisfying $R(X,Y)\overline{C}=0$ is an Einstein manifold and a manifold of scalar curvature $scal=-2n(2n-1)(\alpha^{2}+\beta^{2})$.

Proof.

From equality (3.45), we have $\eta(\overline{C}(X,Y)\xi)=0$ and

[TABLE]

Now,

[TABLE]

[TABLE]

By assumption $R(X,Y)\overline{C}=0$, so we have

[TABLE]

[TABLE]

Therefore

[TABLE]

[TABLE]

From this, it follows that,

[TABLE]

[TABLE]

[TABLE]

Let $\{e_{i}\}$, $i=1,...,2n+1$ be an orthonormal basis. Then summing up for $1\leq i\leq 2n+1$ of the relation (3.48) for $Y=U=e_{i}$ yields

[TABLE]

[TABLE]

Using (3.22) in (3.49), we have

[TABLE]

and $scal=-2n(2n-1)(\alpha^{2}+\beta^{2}).$ ∎

The projective Ricci tensor is defined by

[TABLE]

We have the following

Theorem 3.11**.**

A trans-para-Sasakian manifold satisfying $R(X,Y)\widetilde{P}=0$ is an Einstein manifold and a manifold of scalar curvature $scal=-2n(2n+1)(\alpha^{2}+\beta^{2})$.

Proof.

From the identity $R(X,Y)\widetilde{P}=0$, we get

[TABLE]

Putting $X=U=\xi$ and using (3.15) and (3.52) we have

[TABLE]

Using (3.52) in (3.53), we obtain that $Ric(X,Y)=-2n(\alpha^{2}+\beta^{2})g(X,Y)$ and $scal=2n(2n-1)(\alpha^{2}+\beta^{2})$. ∎

The pseudo-projective curvature tensor is defined by

[TABLE]

[TABLE]

where $a,b$ are constants such that $a,b\neq 0$.

We have the following

Theorem 3.12**.**

If a trans-para-Sasakian manifold is pseudo-projectively flat, then it is an Einstein manifold and a manifold of scalar curvature $scal=-2n(2n+1)(\alpha^{2}+\beta^{2})$.

Proof.

Suppose that $\overline{P}(X,Y)Z=0$, then from (3.54), we get

[TABLE]

[TABLE]

Taking the inner product on both sides of (3.55) by $\xi$, we get

[TABLE]

[TABLE]

Putting $X=\xi$ and using (3.15) and (3.22) in (3.56), we get

[TABLE]

[TABLE]

From the identity (3.57), we obtain that $Ric(X,Y)=-2n(\alpha^{2}+\beta^{2})g(Y,Z)$ and $scal=-2n(2n+1)(\alpha^{2}+\beta^{2})$. ∎

Theorem 3.13**.**

A trans-para-Sasakian manifold is satisfying the relation $R(X,Y)\overline{P}=0$ is an Einstein manifold and a manifold of scalar curvature $scal=-2n(2n+1)(\alpha^{2}+\beta^{2})$.

Proof.

From equality (3.54), we have $\eta(\overline{P}(X,Y)\xi)=0$. Now,

[TABLE]

[TABLE]

By assumption $R(X,Y)\overline{P}=0$, so we have

[TABLE]

[TABLE]

Therefore

[TABLE]

[TABLE]

From this, it follows that,

[TABLE]

[TABLE]

[TABLE]

Let $\{e_{i}\}$, $i=1,...,2n+1$ be an orthonormal basis. Then summing up for $1\leq i\leq 2n+1$ of the relation (3.59) for $Y=U=e_{i}$ yields

[TABLE]

Taking the trace of the identity, we obtain

[TABLE]

From identity (3.61), we get

[TABLE]

Taking $Z=\xi$ in (3.62) and using (3.22) we obtain

[TABLE]

∎

The PC-Bochner curvature tensor on $M$ is defined by [9]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $k=-\frac{scal-2n}{2n+2}$.

Using the PC-Bochner curvature tensor we have

Theorem 3.14**.**

If a trans-para-Sasakian manifold is para-contact conformally flat, then $\alpha^{2}+\beta^{2}=1$.

Proof.

Suppose that the manifold is para-contact conformally flat. Then the condition $\mathbf{B}(X,Y)Z=0$ holds. Putting $X=Z=\xi$ and using (3.16), we obtain

[TABLE]

Since $Y-\eta(Y)\xi=\varphi^{2}Y\neq 0$, we have $\alpha^{2}+\beta^{2}-1=0$. ∎

Theorem 3.15**.**

If a trans-para-Sasakian manifold satisfies the condition $\mathbf{B}(\xi,Y)Ric=0$, then it is either an Einstein manifold with scalar curvature $scal=-2n(2n+1)(\alpha^{2}+\beta^{2})$ or $\alpha^{2}+\beta^{2}=1$.

Proof.

Suppose that the condition $\mathbf{B}(\xi,Y)Ric(Z,V)=0$ holds.This condition implies that

[TABLE]

Putting $V=\xi$ and using (3.16), we obtain

[TABLE]

∎

Acknowledgments

S.Z. is partially supported by Contract DN 12/3/12.12.2017 and Contract 80-10-24/17.04.2018 with the Sofia University St.Kl.Ohridski .