Results on para-Sasakian manifold admitting a quarter symmetric metric connection

TL;DR

This paper investigates the geometric properties of para-Sasakian manifolds with quarter-symmetric metric connections, focusing on pseudosymmetry conditions, and provides explicit examples in 3 and 5 dimensions.

Contribution

It introduces the study of pseudosymmetry conditions on para-Sasakian manifolds with quarter-symmetric metric connections and constructs explicit low-dimensional examples.

Findings

Characterization of pseudosymmetric para-Sasakian manifolds with quarter-symmetric metric connections

Explicit 3D and 5D examples verifying theoretical results

Insights into curvature properties under these connections

Abstract

In this paper we have studied pseudosymmetric, Ricci-pseudosymmetric and projectively pseudosymmetric para-Sasakian manifold admitting a quarter-symmetric metric connection and constructed examples of 3-dimensional and 5-dimensional para-Sasakian manifold admitting a quarter-symmetric metric connection to verify our results.

Equations

$$R(X,Y)\cdot \mathcal{J}=L_{\mathcal{J}}[(X{\wedge }_{g}Y)\cdot \mathcal{J}],$$

$${\varphi }^2=I-\eta o\xi ,\eta (\xi )=1,\varphi (\xi )=0,\eta (\varphi X)=0,$$

$$g(X,\xi )=\eta (X),g(\varphi X,\varphi Y)=g(X,Y)-\eta (X)\eta (Y),$$

$$({\nabla }_X\varphi )Y=-g(X,Y)\xi -\eta (Y)X+2\eta (X)\eta (Y)\xi ,$$

$$d\eta =0and{\nabla }_X\xi =\varphi X,$$

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

$$\eta (R(X,Y)Z)=g(X,Z)\eta (Y)-g(Y,Z)\eta (X),$$

$$R(X,Y)\xi =\eta (X)Y-\eta (Y)X,R(\xi ,X)Y=\eta (Y)X-g(X,Y)\xi ,$$

$$S(X,\xi )=-(n-1)\eta (X),$$

$$S(\varphi X,\varphi Y)=S(X,Y)+(n-1)\eta (X)\eta (Y),$$

$${\tilde{\nabla }}_{X}Y={\nabla }_{X}Y+\eta (Y)\varphi X-g(\varphi X,Y)\xi .$$

$$\tilde{R}(X,Y)Z=R(X,Y)Z+3g(\varphi X,Z)\varphi Y-3g(\varphi Y,Z)\varphi X$$

$$+\{\eta (X)Y-\eta (Y)X\}\eta (Z)-[g(Y,Z)\eta (X)-\eta (Y)g(X,Z)]\xi .$$

$$\tilde{S}(Y,Z)=S(Y,Z)+2g(Y,Z)-(n+1)\eta (Y)\eta (Z)-3trace\varphi g(\varphi Y,Z),$$

$$\tilde{R}(X,Y)\cdot \tilde{R}=L_{\tilde{R}}[(X{\wedge }_{g}Y)\cdot \tilde{R}],$$

$$\tilde{R}(\xi ,Y)\tilde{R}(U,V)W-\tilde{R}(\tilde{R}(\xi ,Y)U,V)W-\tilde{R}(U,\tilde{R}(\xi ,Y)V)W$$

$$-\tilde{R}(U,V)\tilde{R}(\xi ,Y)W=L_{\tilde{R}}[(\xi {\wedge }_{g}Y)\tilde{R}(U,V)W-\tilde{R}((\xi {\wedge }_{g}Y)U,V)W$$

$$-\tilde{R}(U,(\xi {\wedge }_{g}Y)V)W-\tilde{R}(U,V)(\xi {\wedge }_{g}Y)W].$$

$$(L_{\tilde{R}}+2)[\eta (\tilde{R}(U,V)W)Y-g(Y,\tilde{R}(U,V)W)\xi -\eta (U)\tilde{R}(Y,V)W+g(Y,U)\tilde{R}(\xi ,V)W$$

$$-\eta (V)\tilde{R}(U,Y)W+g(Y,V)\tilde{R}(U,\xi )W-\eta (W)\tilde{R}(U,V)Y+g(Y,W)\tilde{R}(U,V)\xi ]=0.$$

$$(L_{\tilde{R}}+2)[g(Y,R(U,V)W)+3g(\varphi U,W)g(\varphi V,Y)-3g(\varphi V,W)g(\varphi U,Y)$$

$$+\eta (W)\{\eta (U)g(V,Y)-\eta (V)g(U,Y)\}-\{g(V,W)\eta (U)-\eta (V)g(U,W)\}\eta (Y)$$

$$+2\{g(V,W)g(Y,U)-g(V,Y)g(U,W)\}]=0.$$

$$g(Y,R(U,V)W)+3g(\varphi U,W)g(\varphi V,Y)-3g(\varphi V,W)g(\varphi U,Y)$$

$$+\eta (W)\{\eta (U)g(V,Y)-\eta (V)g(U,Y)\}-[g(V,W)\eta (U)-\eta (V)g(U,W)]\eta (Y)$$

$$+2[g(V,W)g(Y,U)-g((V,Y)g(U,W)]=0.$$

$$\tilde{S}(Y,U)=-2(n-1)g(Y,U).$$

$$\tilde{R}\cdot \tilde{S}=L_{\tilde{S}}[(X{\wedge }_{g}Y)\cdot \tilde{S}],$$

$$\tilde{S}(\tilde{R}(X,Y)Z,W)+\tilde{S}(Z,\tilde{R}(X,Y)W)=L_{\tilde{S}}[\tilde{S}((X{\wedge }_{g}Y)Z,W)+\tilde{S}(Z,(X{\wedge }_{g}Y)W)].$$

$$(L_{\tilde{S}}+2)[\tilde{S}(X,Z)+2(n-1)g(X,Z)]=0$$

$$P(X,Y)Z=R(X,Y)Z-\frac{1}{(n-1)}[S(Y,Z)X-S(X,Z)Y].$$

$$\tilde{P}(X,Y)Z=\tilde{R}(X,Y)Z-\frac{1}{(n-1)}[\tilde{S}(Y,Z)X-\tilde{S}(X,Z)Y].$$

$$g(\tilde{R}(X,Y)Z,W)=\frac{1}{(n-1)}[\tilde{S}(Y,Z)g(X,W)-\tilde{S}(X,Z)g(Y,W)].$$

$$\tilde{S}(X,Z)=-2(n-1)g(X,Z).$$

$$\tilde{R}(X,Y)\cdot \tilde{P}=L_{\tilde{P}}[(X{\wedge }_{g}Y)\cdot \tilde{P}],$$

$$\tilde{R}(X,\xi )\tilde{P}(U,V)\xi -\tilde{P}(\tilde{R}(X,\xi )U,V)\xi -\tilde{P}(U,\tilde{R}(X,\xi )V)\xi$$

$$-\tilde{P}(U,V)\tilde{R}(X,\xi )\xi =L_{\tilde{P}}[(X{\wedge }_g\xi )\tilde{P}(U,V)\xi -\tilde{P}((X{\wedge }_g\xi )U,V)\xi$$

Full text

Results on para-Sasakian manifold admitting a quarter symmetric metric connection

Vishnuvardhana. S.V., Venkatesha

Abstract: In this paper we have studied pseudosymmetric, Ricci-pseudosymmetric and projectively pseudosymmetric para-Sasakian manifold admitting a quarter-symmetric metric connection and constructed examples of 3-dimensional and 5-dimensional para-Sasakian manifold admitting a quarter-symmetric metric connection to verify our results.

Key Words: Para-Sasakian manifold, pseudosymmetric, Ricci-pseudosymmetric, projectively pseudosymmetric, quarter-symmetric metric connection.

AMS Subject Classification: 53C35,53D40.

1. Introduction

One of the most important geometric property of a space is symmetry. Spaces admitting some sense of symmetry play an important role in differential geometry and general relativity. Cartan [6] introduced locally symmetric spaces, i.e., the Riemannian manifold $(M,g)$ for which $\nabla R=0$, where $\nabla$ denotes the Levi-Civita connection of the metric. The integrability condition of $\nabla R=0$ is $R\cdot R=0$. Thus, every locally symmetric space satisfies $R\cdot R=0$, whereby the first R stands for the curvature operator of $(M,g)$, i.e., for tangent vector fields $X$ and $Y$ one has $R(X,Y)=\nabla_{X}\nabla_{Y}-\nabla_{Y}\nabla_{X}-\nabla_{[X,Y]}$, which acts as a derivation on the second $R$ which stands for the Riemann-Christoffel curvature tensor. The converse however does not hold in general. The spaces for which $R\cdot R=0$ holds at every point were called semi-symmetric spaces and which were classified by Szabo [24].

Semisymmetric manifolds form a subclass of the class of pseudosymmetric manifolds. In some spaces $R\cdot R$ is not identically zero, these turn out to be the pseudo-symmetric spaces of Deszcz [12, 13, 14], which are characterised by the condition $R\cdot R=L\,Q(g,R)$, where $L$ is a real function on $M$ and $Q(g,R)$ is the Tachibana tensor of $M$.

If at every point of $M$ the curvature tensor satisfies the condition

[TABLE]

then a Riemannian manifold $M$ is called pseudosymmetric (resp., Ricci-pseudosymmetric, projectively pseudosymmetric) when $\mathcal{J}=R(resp.,S,P)$ . Here $(X\wedge_{g}Y)$ is an endomorphism and is defined by $(X\wedge_{g}Y)Z=g(Y,Z)X-g(X,Z)Y$ and $L_{\mathcal{J}}$ is some function on $U_{\mathcal{J}}=\{x\in M:\mathcal{J}\neq 0\}$ at $x$. A geometric interpretation of the notion of pseudosymmetry is given in [16]. It is also easy to see that every pseudosymmetric manifold is Ricci-pseudosymmetric, but the converse is not true.

An analogue to the almost contact structure, the notion of almost paracontact structure was introduced by Sato [23]. An almost contact manifold is always odd-dimensional but an almost paracontact manifold could be of even dimension as well. Kaneyuki and Williams [17] studied the almost paracontact structure on a pseudo-Riemannian manifold. Recently, almost paracontact geometry in particular, para-Sasakian geometry has taking interest, because of its interplay with the theory of para-Kahler manifolds and its role in pseudo-Riemannian geometry and mathematical physics ([4, 9, 10], etc.,).

As a generalization of semi-symmetric connection, quarter-symmetric connection was introduced. Quarter-symmetric connection on a differentiable manifold with affine connection was defined and studied by Golab [15]. From thereafter many geometers studied this connection on different manifolds.

Para-Sasakian manifold with respect to quarter-symmetric metric connection was studied by De et.al., [[19, 2]], Pradeep Kumar et.al., [21] and Bisht and Shanker [18].

Motivated by the above studies in this article we study properties of projective curvature tensor on para-Sasakian manifold admitting a quarter-symmetric metric connection. The organization of the paper is as follows: In Section 2, we present some basic notions of para-Sasakian manifold and quarter-symmetric metric connection on it. Section 3 and 4 are respectively devoted to study the pseudosymmetric and Ricci-pseudosymmetric para-Sasakian manifold admitting a quarter-symmetric metric connection. Here we prove that if a para-Sasakian manifold $M^{n}$ admitting a quarter-symmetric metric connection is Pseudosymmetric (resp., Ricci pseudosymmetric) then $M^{n}$ is an Einstein manifold with respect to quarter-symmetric metric connection or it satisfies $L_{\tilde{R}}=-2$ (resp., $L_{\tilde{S}}=-2$). Section 5 and 6 are concerned with projectively flat and projectively pseudosymmetric para-Sasakian manifold $M^{n}$ admitting a quarter-symmetric metric connection. Finally, we construct examples of 3-dimensional and 5-dimensional para-Sasakian manifold admitting a quarter-symmetric metric connection and we find some of its geometric characteristics.

2. Preliminaries

A differential manifold $M^{n}$ is said to admit an almost paracontact Riemannian structure $(\phi,\xi,\eta,g)$, where $\phi$ is a tensor field of type (1, 1), $\xi$ is a vector field, $\eta$ is a 1-form and $g$ is a Riemannian metric on $M^{n}$ such that

[TABLE]

for all vector fields $X,Y\in T(M^{n})$. If $(\phi,\xi,\eta,g)$ on $M^{n}$ satisfies the following equations

[TABLE]

then $M^{n}$ is called para-Sasakian manifold [3].

In a para-Sasakian manifold, the following relations hold [7]:

[TABLE]

[TABLE]

for every vector fields $X,Y,Z$ on $M^{n}$.

Here we consider a quarter-symmetric metric connection $\tilde{\nabla}$ on a para-Sasakian manifold [19] given by

[TABLE]

The relation between curvature tensor $\tilde{R}(X,Y)Z$ of $M^{n}$ with respect to quarter-symmetric metric connection $\tilde{\nabla}$ and the curvature tensor $R(X,Y)Z$ with respect to the Levi-Civita connection $\nabla$ is given by

[TABLE]

Also from (2.11) we obtain

[TABLE]

where $\tilde{S}$ and $S$ are Ricci tensors of connections $\tilde{\nabla}$ and $\nabla$ respectively.

3. Pseudosymmetric para-Sasakian manifold admitting a quarter-symmetric metric connection

A para-Sasakian manifold $M^{n}$ admitting a quarter-symmetric metric connection is said to be pseudosymmetric if

[TABLE]

holds on the set $U_{\tilde{R}}=\{x\in M^{n}:\tilde{R}\neq 0\,\,at\,\,x$}, where $L_{\tilde{R}}$ is some function on $U_{\tilde{R}}$.

Suppose that $M^{n}$ be pseudosymmetric, then in view of (3.1) we have

[TABLE]

By virtue of (2.7) and (2.11), (3.2) takes the form

[TABLE]

Taking inner product of (3.3) with $\xi$ and using (2.6) and (2.11), we get

[TABLE]

Assuming that $L_{\tilde{R}}+2\neq 0$, the above equation becomes

[TABLE]

Putting $V=W=e_{i}$, where $\{e_{i}\}$ is an orthonormal basis of the tangent space at each point of the manifold and taking summation over $i$, $i=1,2,3,\cdots,n$, we get

[TABLE]

Hence, we can state the following:

Theorem 3.1**.**

If a para-Sasakian manifold $M^{n}$ admitting a quarter-symmetric metric connection is pseudosymmetric then $M^{n}$ is an Einstein manifold with respect to quarter-symmetric metric connection or it satisfies $L_{\tilde{R}}=-2$.

4. Ricci-pseudosymmetric para-Sasakian manifold admitting a quarter-symmetric metric connection

A para-Sasakian manifold $M^{n}$ admitting a quarter-symmetric metric connection is said to be Ricci-pseudosymmetric if the following condition is satisfied

[TABLE]

on $U_{\tilde{S}}$.

Let para-Sasakian manifold $M^{n}$ admitting a quarter-symmetric metric connection be Ricci-pseudosymmetric. Then we have

[TABLE]

By taking $Y=W=\xi$ and making use of (2.7), (2.8) and (2.11), the above equation turns into

[TABLE]

Thus, we have the following assertion:

Theorem 4.2**.**

If a para-Sasakian manifold $M^{n}$ admitting a quarter-symmetric metric connection is Ricci pseudosymmetric then $M^{n}$ is an Einstein manifold with respect to quarter-symmetric metric connection or it satisfies $L_{\tilde{S}}=-2$.

5. Projectively flat para-Sasakian manifold admitting a quarter-symmetric metric connection

The projective curvature tensor on a Riemannian manifold is defined by

[TABLE]

For an $n$-dimensional para-Sasakian manifold $M^{n}$ admitting a quarter-symmetric metric connection, the projective curvature tensor is given by

[TABLE]

Theorem 5.3**.**

A projectively flat para-Sasakian manifold $M^{n}$ admitting a quarter-symmetric metric connection is an Einstein manifold with respect to quarter-symmetric metric connection.

Proof.

Consider a projectively flat para-Sasakian manifold admitting a quarter-symmetric metric connection. Then from (5.2) we have

[TABLE]

Setting $X=W=\xi$ in (5.3) and using (2.7), (2.8), (2.11) and (2.12), we get

[TABLE]

Hence the proof. ∎

6. Projectively pseudosymmetric para-Sasakian manifold admitting a quarter-symmetric metric connection

A para-Sasakian manifold admitting a quarter-symmetric metric connection is said to be projectively pseudosymmetric if

[TABLE]

holds on the set $U_{\tilde{P}}=\{x\in M^{n}:\tilde{P}\neq 0\}$ at $x$, where $L_{\tilde{P}}$ is some function on $U_{\tilde{P}}$.

Let $M^{n}$ be projectively pseudosymmetric, then we have

[TABLE]

By virtue of (2.11), (2.12) and (5.2), (6.2) becomes

[TABLE]

So, one can state that:

Theorem 6.4**.**

If a para-Sasakian manifold $M^{n}$ admitting a quarter-symmetric metric connection is projectively pseudosymmetric then $M^{n}$ is projectively flat with respect to quarter-symmetric metric connection or $L_{\tilde{P}}=-2$.

In view of theorem 5.3, one can state the above theorem as

Theorem 6.5**.**

If a para-Sasakian manifold $M^{n}$ admitting a quarter-symmetric metric connection is projectively pseudosymmetric then $M^{n}$ is an Einstein manifold with respect to quarter-symmetric metric connection or $L_{\tilde{P}}=-2$.

7. **Example **

7.1. Example

We consider a 3-dimensional manifold $M=\{(x,y,z)\in\mathbb{R}^{3}:z\neq 0\}$, where $(x,y,z)$ are standard coordinates in $\mathbb{R}^{3}$. Let $\{E_{1},E_{2},E_{3}\}$ be a linearly independent global frame field on $M$ given by

[TABLE]

If $g$ is a Riemannian metric defined by

[TABLE]

for $1\leq i,j\leq 3,$ and if $\eta$ is the 1-form defined by $\eta(Z)=g(Z,E_{3})$ for any vector field $Z\in\chi(M)$. We define the (1, 1)-tensor field $\phi$ as

[TABLE]

The linearity property of $\phi$ and $g$ yields that

[TABLE]

for any $U,V\in\chi(M)$.

Now we have

[TABLE]

The Riemannian connection $\nabla$ of the metric $g$ known as Koszul’s formula and is given by

[TABLE]

Using Koszul’s formula we get the followings in matrix form

[TABLE]

Clearly $(\phi,\xi,\eta,g)$ is a para-Sasakian structure on $M$. Thus $M(\phi,\xi,\eta,g)$ is a 3-dimensional para-Sasakian manifold.

Using (2.10) and the above equation, one can easily obtain the following:

[TABLE]

With the help of the above results it can be easily verified that

[TABLE]

and

[TABLE]

Since ${E_{1},E_{2},E_{3}}$ forms a basis, any vector field $X,Y,Z\in\chi(M)$ can be written as $X=a_{1}E_{1}+b_{1}E_{2}+c_{1}E_{3}$, $Y=a_{2}E_{1}+b_{2}E_{2}+c_{2}E_{3}$, $Z=a_{3}E_{1}+b_{3}E_{2}+c_{3}E_{3}$, where $a_{i},b_{i},c_{i}\in\mathbb{R}^{+}$ (the set of all positive real numbers), $i=1,2,3$. Using the expressions of the curvature tensors, we find values of Riemannian curvature and Ricci curvature with respect to quarter-symmetric metric connection as;

[TABLE]

Using (7.1), (7.3) and the expression of the endomorphism $(X\wedge_{g}Y)Z=g(Y,Z)X-g(X,Z)Y$, one can easily verify that

[TABLE]

here $L_{\tilde{S}}=-2$. Thus, the above equation verify one part of the theorem 4.2 of section 4

Moreover, the manifold under consideration satisfies

[TABLE]

Hence, from the above equations one can say that this example verifies the condition $c$ of theorem 3.1 in [2] and first Bianchi identity.

7.2. Example

We consider a 5-dimensional manifold $M=\{(x_{1},x_{2},x_{3},x_{4},x_{5})\in\mathbb{R}^{5}\}$, where $(x_{1},x_{2},x_{3},x_{4},x_{5})$ are standard coordinates in $\mathbb{R}^{5}$. We choose the vector fields

[TABLE]

which are linearly independent at each point of $M$

Let $g$ be a Riemannian metric defined by

[TABLE]

for $1\leq i,j\leq 5,$ and if $\eta$ is the 1-form defined by $\eta(Z)=g(Z,E_{5})$ for any vector field $Z\in\chi(M)$. Let $\phi$ be the (1, 1)-tensor field defined by

[TABLE]

The linearity property of $\phi$ and $g$ yields that

[TABLE]

for any $U,V\in\chi(M)$.

Now we have

[TABLE]

By virtue of Koszul’s formula we get the followings in matrix form

[TABLE]

Above expressions satisfies all the properties of para-Sasakian manifold. Thus $M(\phi,\xi,\eta,g)$ is a 5-dimensional para-Sasakian manifold.

From the above expressions and the relation of quarter symmetric metric connection and Riemannian connection, one can easily obtain the following:

[TABLE]

With the help of the above results it can be easily obtain the non-zero components of curvature tensors as

[TABLE]

and

[TABLE]

Since ${E_{1},E_{2},E_{3},E_{4},E_{5}}$ forms a basis, any vector field $X,Y,Z\in\chi(M)$ can be written as $X=a_{1}E_{1}+b_{1}E_{2}+c_{1}E_{3}+d_{1}E_{4}+f_{1}E_{5}$, $Y=a_{2}E_{1}+b_{2}E_{2}+c_{2}E_{3}+d_{2}E_{4}+f_{2}E_{5}$, $Z=a_{3}E_{1}+b_{3}E_{2}+c_{3}E_{3}+d_{3}E_{4}+f_{3}E_{5}$, where $a_{i},b_{i},c_{i},d_{i},f_{i}\in\mathbb{R}^{+}$ (the set of all positive real numbers), $i=1,2,3,4,5$. Using the expressions of the curvature tensors, we find values of Riemannian curvature and Ricci curvature with respect to quarter-symmetric metric connection as;

[TABLE]

In view of (7.5), (7.6) and the expression of the endomorphism one can easily verify the equation (7.4) and hence the theorem 4.2 of section 4 is verified. This example also verifies the condition $c$ of theorem 3.1 in [2] and first Bianchi identity.

Above two examples verifies the one part of the theorem 4.2, that is, if a para-Sasakian manifold $M^{n}$ admitting a quarter-symmetric metric connection is Ricci pseudosymmetric then $M^{n}$ satisfies $L_{\tilde{S}}=-2$ ($M^{n}$ is not Einstein manifold with respect to quarter-symmetric metric connection). Another part of the theorem is that, if a para-Sasakian manifold $M^{n}$ admitting a quarter-symmetric metric connection is Ricci pseudosymmetric then $M^{n}$ is an Einstein manifold with respect to quarter-symmetric metric connection ($L_{\tilde{S}}\neq-2$). Now, the second part of the theorem 4.2 can be verified by using the proper example.

7.3. Example

We consider a 5-dimensional manifold $M=\{(x,y,z,u,v)\in\mathbb{R}^{5}\}$, where $(x,y,z,u,v)$ are standard coordinates in $\mathbb{R}^{5}$. Let $\{E_{1},E_{2},E_{3},E_{4},E_{5}\}$ be a linearly independent global frame field on $M$ given by

[TABLE]

Let $g$ be a Riemannian metric defined by

[TABLE]

for $1\leq i,j\leq 5,$ and if $\eta$ is the 1-form defined by $\eta(Z)=g(Z,E_{1})$ for any vector field $Z\in\chi(M)$. Let the (1, 1)-tensor field $\phi$ be defined by

[TABLE]

With the help of linearity property of $\phi$ and $g$, we have

[TABLE]

for any $X,Y\in\chi(M)$.

Now we have

[TABLE]

With the help of Koszul’s formula we get the followings in matrix form

[TABLE]

In this case, $(\phi,\xi,\eta,g)$ is a para-Sasakian structure on $M$ and hence $M(\phi,\xi,\eta,g)$ is a 5-dimensional para-Sasakian manifold.

Using (2.10) and the above equation, one can easily obtain the following:

[TABLE]

From above results it can be easily obtain the non-zero components of Riemannian curvature and Ricci curvature tensors as

[TABLE]

and

[TABLE]

where $X=a_{1}E_{1}+b_{1}E_{2}+c_{1}E_{3}+d_{1}E_{4}+f_{1}E_{5}$ and $Y=a_{2}E_{1}+b_{2}E_{2}+c_{2}E_{3}+d_{2}E_{4}+f_{2}E_{5}$.

From (7.7), (7.8) and the expression of the endomorphism one can easily substantiate, the equation (7.4) and hence second part of the theorem 4.2 (for $L_{\tilde{S}}\neq-2$).