Majority of present approaches to mathematical general relativity launch with the concept of a manifold. The standpoint of physics and relativity is to the investigation of manifolds with indefinite metrics. Several authors have studied manifold with indefinite matrices. Bejancu and Duggal [1] originated the concept of ò-Sasakian manifolds in 1993. De and Sarkar [2] pioneered (ò)-Kenmotsu manifolds and investigated some curvature conditions on it. Pandey and Tiwari [3] constructed the relation between semi-symmetric metric connection and Riemannian connection of (ò)-Kenmotsu manifolds and have studied several curvature conditions. The notion of (ò)-Para Sasakian Manifolds was pioneered by Tripathi et al. [4] in 2009.

The Riemannian symmetric spaces were introduced by French mathematician Carton during the nineteenth century and play a main tool in differential geometry. A Riemannian manifold is locally symmetric [5] , if ∇ R = 0 , where R is the Riemannian curvature tensor of ( M , g ) . During the last five decades the notion of locally symmetric manifolds has been studied by many authors in several ways to a different extent such as recurrent manifold by Walker [6] , semisymmetric manifold by Szabó [7] , pseudosymmetric manifold in the sense of Deszcz [8] , a non-flat Riemannian manifold ( M n , g ) ( n > 2 ) is said to be pseudosymmetric in the sense of Chaki [9] if it satisfies the relation

( ∇ W R ) ( X , Y , Z , U ) = 2 A ( W ) R ( X , Y , Z , U ) + A ( X ) R ( W , Y , Z , U ) + A ( Y ) R ( X , W , Z , U )       + A ( Z ) R ( X , Y , W , U ) + A ( U ) R ( X , Y , Z , W ) , (1.1)

i.e.,

( ∇ W R ) ( X , Y ) Z = 2 A ( W ) R ( X , Y ) Z + A ( X ) R ( W , Y ) Z + A ( Y ) R ( X , W ) Z       + A ( Z ) R ( X , Y ) W + g ( R ( X , Y ) Z , W ) ρ , (1.2)

for any X , Y , Z , U , W ∈ T P ( M ) and where R is the Riemannian curvature tensor of the manifold, A is a non-zero 1-form such that g ( X , ρ ) = A ( X ) for every vector field X. Every recurrent manifold is pseudosymmetric in the sense of Chaki [9] but not conversely. The pseudosymmetry in the sense of Chaki is not equivalent to that in the sense of Deszcz [8] . However, the pseudosymmetry by Chaki will be the pseudosymmetry by Deszcz if and only if the non-zero 1-form associated with n-dimensional pseudosymmetry is closed. Pseudosymmetric manifolds also have been studied by Chaki and De [10] , Özen and Altay [11] , Tarafder [12] , De, Murathan and Özgür [13] , Tarafder and De [14] and others. Many authors have been weakened by Ricci symmetry that has been differently extended such as a Ricci recurrent, Ricci symmetric and pseudo Ricci symmetric for past two decades.

A non-flat Riemannian manifold ( M n , g ) is said to be pseudo-Ricci symmetric [15] if its Ricci tensor S of type ( 0,2 ) is not identically zero and satisfies the condition

( ∇ X S ) ( Y , Z ) = 2 A ( X ) S ( Y , Z ) + A ( Y ) S ( X , Z ) + A ( Z ) S ( Y , X ) , (1.3)

for any X , Y , Z ∈ T P ( M ) where A is a nowhere vanishing 1-form and ∇ refers the operator of covariant differentiation with respect to the metric tensor g. Such a n-dimensional manifold is denoted by ( P R S ) n . The pseudo-Ricci symmetric manifolds have also been studied by Arslan et al. [16] , De and Mazumder [17] and many others. The notion of locally ϕ-symmetric Sasakian manifold was introduced by Takahashi [18] due to a weaker version of locally symmetry. Generating the notion of locally ϕ-symmetric Sasakian manifolds, De et al. [19] , introduce the notion of ϕ-recurrent Sasakian manifolds also Shukla et al. [20] studied ϕ-symmetric and ϕ Ricci symmetric para Sasakian manifolds.

Inspired by above studies this paper makes an attempt to study of ϕ-pseudo symmetric and ϕ-pseudo Ricci symmetric ò-para Sasakian manifolds. It is organized as follows. Section 2 is related with ò-para Sasakian manifolds. Section 3 is dealt with the study of ϕ pseudo symmetric ò-para Sasakian manifolds. In Section 4, we study of ϕ-pseudo Concircularly symmetric ò-para Sasakian manifold. In Section 5, we study ϕ-pseudo Ricci symmetric ò-para Sasakian manifold. The relation (1.3) can be written as

( ∇ X Q ) ( Y ) = 2 A ( X ) Q ( Y ) + A ( Y ) Q ( X ) + S ( Y , X ) ρ , (1.4)

where ρ is the vector field associated to the 1-form A such that A ( X ) = g ( X , ρ ) and Q is the Ricci operator, i.e., g ( Q X , Y ) = S ( X , Y ) .

Let ( M n , g ) be an almost paracontact manifold is equipped with an almost paracontact structure ( ϕ , ξ , η ) consisting of a tensor field ϕ of type ( 1,1 ) , a vector field ξ and a 1-form η satisfying

ϕ 2 X = X − η ( X ) ξ , (2.1)

η ( ξ ) = − 1 ,         ϕ ξ = 0 ,           η ∘ ϕ = 0 , (2.2)

g ( ϕ X , ϕ Y ) = g ( X , Y ) − ϵ η ( X ) η ( Y ) , (2.3)

where ϵ = ± 1 , in this case ( M n , g ) is called an (ò)-almost paracontact metric manifold equipped with an (ò)-almost paracontact structure ( ϕ , ξ , η , g ) [21] . In particular, index ( g ) = 1 , then (ò)-almost paracontact metric manifold will be called a Lorentzian almost paracontact metric manifold. In view of equation [22] [23] , we have

g ( ϕ X , Y ) = g ( X , ϕ Y ) , (2.4)

g ( x , ξ ) = ϵ η ( X ) , (2.5)

for any X , Y ∈ T p M , the structure of a vector field ξ is a never light like. An (ò)-almost paracontact metric manifold (respectively a Lorentzian almost paracontact manifold ( M n , ϕ , ξ , g , ϵ ) is said to be space-like (ò)-almost paracontact metric manifold (respectively a space-like Lorentzian almost paracontact manifold), if ϵ = 1 and ( M n , g ) is said to be a time-like (ò)-almost paracontact metric manifold (respectively a Lorentzian almost paracontact manifold), if ϵ = − 1 . An (ò)-almost paracontact metric structure is called an (ò)-Para Sasakian structure if

( ∇ X ϕ ) ( Y ) = − g ( X , ϕ Y ) ξ − ϵ η ( Y ) ϕ 2 X , (2.6)

where ∇ is the Levi-Civita connection. A manifold ( M n , g ) endowed with an (ò)-para Sasakian structure is called an (ò)-para Sasakian manifold. For ϵ = 1 and g is a Riemannian, ( M n , g ) is the usual para Sasakian manifold [24] . For ϵ = − 1 , g Lorentzian and ξ replaced by − ξ , ( M n , g ) becomes a Lorentzian para Sasakian manifold [23] . In an (ò)-para Sasakian manifold, we have

∇ X ξ = ϵ ϕ X , (2.7)

g ( ξ , ξ ) = ± 1 = ϵ , (2.8)

( ∇ X η ) ( Y ) = ϵ g ( ϕ X , Y ) = Ω ( X , Y ) , (2.9)

for any X , Y ∈ T p M , where Ω is the fundamental 2-form. In an (ò)-almost para Sasakian manifold ( M n , g ) , the following relations are hold.

η ( R ( X , Y ) Z ) = ϵ [ g ( Y , Z ) η ( X ) − g ( X , Z ) η ( Y ) ] , (2.10)

R ( ξ , X ) Y = − ϵ g ( X , Y ) ξ − ϵ η ( Y ) X , (2.11)

R ( X , Y ) ξ = − ϵ η ( Y ) X + ϵ η ( X ) Y , (2.12)

( ∇ X R ) ( Y , Z ) ξ = ϵ 2 [ g ( ϕ X , Y ) Z − g ( ϕ X , Z ) Y ] . (2.13)

In an n-dimensional (ò)-para Sasakian manifold ( M n , g ) , the Ricci tensor satisfies

S ( ϕ X , ϕ Y ) = S ( X , Y ) + ( n − 1 ) η ( X ) η ( Y ) , (2.14)

S ( X , ξ ) = S ( ξ , X ) = − ( n − 1 ) η ( X ) . (2.15)

Definition 3.1. A ò-Para Sasakian manifold ( M n ) ( ϕ , ξ , η , g ) is said to be a ϕ-pseudo symmetric if the curvature tensor R satisfies

ϕ 2 ( ( ∇ W R ) ( X , Y ) Z ) = 2 A ( W ) R ( X , Y ) Z + A ( X ) R ( W , Y ) Z + A ( Y ) R ( X , W ) Z       + A ( Z ) R ( X , Y ) W + g ( R ( X , Y ) Z , W ) ρ , (3.1)

for any X , Y , Z , W ∈ T P M . If A = 0 the manifold is said to be ϕ-symmetric.

By virtue of (2.1), it follows that

( ∇ W R ) ( X , Y ) Z − η ( ( ∇ W R ) ( X , Y ) Z ) ξ = 2 A ( W ) R ( X , Y ) Z + A ( X ) R ( W , Y ) Z + A ( Y ) R ( X , W ) Z       + A ( Z ) R ( X , Y ) W + g ( R ( X , Y ) Z , W ) ρ , (3.2)

from which it follows that

g ( ( ∇ W R ) ( X , Y ) Z , U ) − η ( ( ∇ W R ) ( X , Y ) Z ) η ( U ) = 2 A ( W ) g ( R ( X , Y ) Z , U ) + A ( X ) g ( R ( W , Y ) Z , U )       + A ( Y ) g ( R ( X , W ) Z , U ) + A ( Z ) g ( R ( X , Y ) W , U )       + g ( R ( X , Y ) Z , W ) A ( U ) . (3.3)

Taking an orthonormal frame field and contracting (3.3) over X and U, then by using (2.2) and (2.5), we get

( ∇ W S ) ( Y , Z ) − g ( ( ∇ W R ) ( ξ , Y ) Z , ξ ) = 2 A ( W ) S ( Y , Z ) + A ( Y ) S ( W , Z ) + A ( Z ) S ( Y , W )       + A ( R ( W , Y ) Z ) + A ( R ( W , Z ) Y ) . (3.4)

Using (2.11) and (2.13), we have

g ( ( ∇ W R ) ( ξ , Y ) Z , ξ ) = 0 , (3.5)

by virtue of (3.5), it follows from (3.4) that

( ∇ W S ) ( Y , Z ) = 2 A ( W ) S ( Y , Z ) + A ( Y ) S ( W , Z ) + A ( Z ) S ( Y , W )     + A ( R ( W , Y ) Z ) + A ( R ( W , Z ) Y ) , (3.6)

This leads to the following:

Theorem 3.1. A ϕ-pseudo symmetric on a ò-para Sasakian manifold is Pseudo-Ricci symmetric if and only if A ( R ( W , Y ) Z ) + A ( R ( W , Z ) Y ) = 0 .

Putting Z = ξ in (3.2), by using (2.10), (2.12) and (2.13), we have

A ( ξ ) R ( X , Y ) W = ϵ 2 [ g ( ϕ W , Y ) X − g ( ϕ W , X ) Y ] + ϵ { 2 A ( W ) [ η ( Y ) X − η ( X ) Y ]       + A ( X ) [ η ( Y ) W − η ( W ) Y ] + A ( Y ) [ η ( X ) W − η ( W ) X ]       + [ η ( Y ) g ( X , W ) − η ( X ) g ( Y , W ) ] ρ } . (3.7)

This leads to the following:

Theorem 3.2. A ϕ-pseudo symmetric on a ò-para Sasakian manifold, the curvature tensor satisfies the relation (3.7).

From (3.7) follows that

A ( ξ ) S ( Y , W ) = ϵ 2 ( n − 1 ) Ω ( W , Y ) + ϵ { ( n − 1 ) η ( Y ) A ( W )       + ( n − 1 ) η ( W ) A ( Y ) + η ( Y ) η ( W ) + g ( Y , W ) } , (3.8)

replacing Y by ϕ Y and W by ϕ W and using (2.3), (2.14), we have

S ( Y , W ) = 1 A ( ξ ) { ϵ g ( Y , W ) − [ ϵ 2 + ( n − 1 ) A ( ξ ) ] η ( Y ) η ( W ) + ϵ 2 ( n − 1 ) Ω ( Y , W ) } .

(3.9)

Hence we can state the following:

Theorem 3.3. A ϕ-pseudo symmetric on a ò-para Sasakian manifold, the curvature tensor satisfies the relation (3.9), provided A ( ξ ) ≠ 0 .

Definition 4.2. A n-dimensional ò-para Sasakian manifold is said to be ϕ-pseudo Concircularly symmetric, if its Concircular curvature tensor C ˜ is given by [25]

C ˜ ( X , Y ) Z = R ( X , Y ) Z − r n ( n − 1 ) [ g ( Y , Z ) X − g ( X , Z ) Y ] . (4.1)

Satisfies the relation

ϕ 2 ( ( ∇ W C ˜ ) ( X , Y ) Z ) = 2 A ( W ) C ˜ ( X , Y ) Z + A ( X ) C ˜ ( W , Y ) Z + A ( Y ) C ˜ ( X , W ) Z       + A ( Z ) C ˜ ( X , Y ) W + g ( C ˜ ( X , Y ) Z , W ) ρ , (4.2)

for any X , Y , Z , W ∈ T P M , where A is a non-zero 1-forms, such that A ( X ) = g ( X , ρ ) .

by virtue of (2.1), it follows from (4.2)

( ∇ W C ˜ ) ( X , Y ) Z − η ( ( ∇ W C ˜ ) ( X , Y ) Z ) ξ = 2 A ( W ) C ˜ ( X , Y ) Z + A ( X ) C ˜ ( W , Y ) Z + A ( Y ) C ˜ ( X , W ) Z       + A ( Z ) C ˜ ( X , Y ) W + g ( C ˜ ( X , Y ) Z , W ) ρ , (4.3)

which follows that

g ( ( ∇ W C ˜ ) ( X , Y ) Z , U ) − η ( ( ∇ W C ˜ ) ( X , Y ) Z ) η ( U ) = 2 A ( W ) g ( C ˜ ( X , Y ) Z , U ) + A ( X ) g ( C ˜ ( W , Y ) Z , U )       + A ( Y ) g ( C ˜ ( X , W ) Z , U ) + A ( Z ) g ( C ˜ ( X , Y ) W , U )       + g ( C ˜ ( X , Y ) Z , W ) A ( U ) . (4.4)

Taking an orthonormal frame field and contracting (4.4) over X and U, by using (2.1) and (4.1), we get

( ∇ X S ) ( Y , Z ) − d r ( W ) n g ( Y , Z ) + g ( ( ∇ W C ˜ ) ( ξ , Y ) Z , ξ ) = 2 A ( W ) S ( Y , Z ) + A ( Y ) S ( W , Z ) + A ( Z ) S ( Y , W ) − r n [ 2 A ( W ) g ( Y , Z )       + A ( Y ) g ( W , Z ) + A ( Z ) g ( Y , W ) ] + A ( C ˜ ( W , Y ) Z ) + A ( C ˜ ( W , Z ) Y ) , (4.5)

by virtue of (3.5) and from (4.1), yields

g ( ( ∇ W C ˜ ) ( ξ , Y ) Z , ξ ) = − d r ( W ) n ( n − 1 ) [ g ( Y , Z ) − η ( Y ) η ( Z ) ] . (4.6)

In view of (4.6) from (4.5), we have

( ∇ X S ) ( Y , Z ) − d r ( W ) n g ( Y , Z ) − d r ( W ) n ( n − 1 ) [ g ( Y , Z ) − η ( Y ) η ( Z ) ] = 2 A ( W ) S ( Y , Z ) + A ( Y ) S ( W , Z ) + A ( Z ) S ( Y , W ) − r n [ 2 A ( W ) g ( Y , Z )       + A ( Y ) g ( W , Z ) + A ( Z ) g ( Y , W ) ] + A ( C ˜ ( W , Y ) Z ) + A ( C ˜ ( W , Z ) Y ) . (4.7)

This leads to the following:

Theorem 4.4. A ϕ-pseudo Concircularly symmetric ò-para Sasakian manifold is pseudo-Ricci symmetric if and only if

d r ( W ) n g ( Y , Z ) + d r ( W ) n ( n − 1 ) [ g ( Y , Z ) − η ( Y ) η ( Z ) ] − r n [ 2 A ( W ) g ( Y , Z ) + A ( Y ) g ( W , Z ) + A ( Z ) g ( Y , W ) ] + A ( C ˜ ( W , Y ) Z ) + A ( C ˜ ( W , Z ) Y ) = 0. (4.8)

Putting Z = ξ in (4.3) and using (2.10), (2.12), (2.15) and (4.1), we obtain

ϵ 2 [ Ω ( W , X ) Y − Ω ( W , Y ) X ] − ϵ d r ( W ) n ( n − 1 ) [ η ( Y ) X − η ( X ) Y ]   + ϵ [ 1 − r n ( n − 1 ) ] [ η ( Y ) g ( X , W ) − η ( X ) g ( Y , W ) ] ρ r n ( n − 1 ) A ( ξ ) [ g ( Y , W ) X − g ( X , W ) Y ]   + ϵ [ 1 − r n ( n − 1 ) ] { 2 A ( W ) [ η ( Y ) X − η ( X ) Y ]   + A ( X ) [ η ( Y ) W − η ( W ) Y ] + A ( Y ) [ η ( W ) X − η ( X ) W ] } = A ( ξ ) R ( X , Y ) W . (4.9)

Hence we can state the following:

Theorem 4.5. In a ϕ-pseudo Concircularly symmetric ò-para Sasakian manifold, the curvature tensor satisfies the relation (4.9).

Next, we take inner product of (4.9) with U and taking an orthonormal frame field and contracting (4.9) over X and U, yields

A ( ξ ) S ( Y , W ) = ϵ 2 ( 1 − n ) Ω ( Y , W ) − ϵ d r ( W ) n η ( Y )     + ϵ [ 1 − r n ( n − 1 ) ] [ η ( Y ) η ( W ) − g ( Y , W ) ] + r n A ( ξ ) g ( Y , W )     + ϵ [ 1 − r n ( n − 1 ) ] ( n − 1 ) [ 2 A ( W ) η ( Y ) + A ( Y ) η ( W ) ] . (4.10)

Replacing Y by ϕ Y and W by ϕ W , we obtain

S ( Y , W ) = ϵ 2 ( 1 − n ) A ( ξ ) Ω ( Y , W ) + [ r n − ϵ A ( ξ ) [ 1 − r n ( n − 1 ) ] g ( Y , W ) ]     + [ ϵ 2 A ( ξ ) [ 1 − r n ( n − 1 ) ] − ϵ A ( ξ ) − ( n − 1 ) ] η ( Y ) η ( W ) . (4.11)

This leads to the following:

Theorem 4.6. A ϕ-pseudo Concircularly symmetric ò-para Sasakian manifold, the curvature tensor satisfies the relation (4.11).

Definition 5.3. A n-dimensional ò-para Sasakian manifold is said to be ϕ-pseudo Ricci symmetric, if the Ricci operator Q satisfies

ϕ 2 ( ( ∇ W Q ) ( Y ) ) = 2 A ( X ) Q Y + A ( Y ) Q X + S ( Y , X ) ρ , (5.1)

for any X , Y ∈ T P M , where A is a non zero 1-form.

In particular if A = 0 , then (5.1) turns into ϕ-Ricci symmetric ò-para Sasakian manifold.

In view of (2.1), then (5.1) becomes

( ∇ W Q ) ( Y ) − η ( ( ∇ W Q ) ( Y ) ) ξ = 2 A ( X ) Q Y + A ( Y ) Q X + S ( Y , X ) ρ , (5.2)

which follows

g ( ( ∇ W Q ) ( Y ) , Z ) − S ( ∇ W Y , Z ) − η ( ( ∇ W Q ) ( Y ) ) η ( Z ) = 2 A ( X ) S ( Y , Z ) + A ( Y ) S ( X , Z ) + S ( Y , X ) A ( Z η ) , (5.3)

putting Y = ξ in (5.3), using (2.7) and (2.15), we get

A ( ξ ) S ( X , Z ) + ϵ S ( ϕ X , Z ) = ( n − 1 ) [ ϵ g ( ϕ X , Z ) + 2 A ( X ) η ( Z ) + η ( X ) A ( Z ) ] . (5.4)

Replacing X by ϕ X , Z by ϕ Z in (5.4) and using (2.14), we get

S ( X , Z ) = ϵ A ( ξ ) [ ( n − 1 ) Ω ( X , Z ) − S ( ϕ X , Z ) ] − ( n − 1 ) η ( X ) η ( Z ) . (5.5)

This leads to the following:

Theorem 5.7. A ϕ-pseudo Ricci symmetric ò-para Sasakian manifold, the curvature tensor satisfies the relation (5.5).

The authors declare no conflicts of interest regarding the publication of this paper.

Somashekhara, P. and Venkatesha (2019) ϕ-Pseudo Symmetric ò-Para Sasakian Manifolds. Open Access Library Journal, 6: e5273. https://doi.org/10.4236/oalib.1105273