On Indefinite Almost Paracontact Metric Manifold

Transcription

1 International Mathematical Forum, Vol. 6, 2011, no. 22, On Indefinite Almost Paracontact Metric Manifold K. P. Pandey Department of Applied Mathematics Madhav Proudyogiki Mahavidyalaya Bhopal, (M.P.), India Brajendra Tiwari Department of Mathematics Radharaman Institute of Science and Technology Bhopal, (M.P.), India Abstract The object of the present paper is to introduce special ( ) para Sasakian manifold and to study some of its properties. We have shown that a recurrent special ( ) para Sasakian manifold with η as 1-form of recurrence is Einstein manifold. Among others, we have obtained a necessary and sufficient condition for special ( ) para Sasakian manifold to be quasi-projectively flat and shown that quasi-projectively flat special ( ) para Sasakian manifold is Einstein. Keywords: Indefinite almost paracontact metric manifold, ( ) para Sasakian manifold, 2-Killing vector fields, quasi-projectively flat manifold 1. Introduction The notion of almost paracontact manifold was introduced by Sato [4] in The structure is an analogue of the almost contact structure [3,12] and is closely related to almost product structure [in contrast to almost contact structure, which is related to almost complex structure]. An almost contact manifold is always odd dimensional but an almost paracontact manifold could be even dimensional as well. In 1969, Takahashi [13] introduced almost contact manifold equipped with associated pseudo-riemannian metrics. In particular, he studied Sasakian manifolds equipped with an associated pseudo-riemannian metric. These indefinite almost contact metric manifold and indefinite Sasakian manifolds

2 1072 K. P. Pandey and B. Tiwari are also known as ( ) almost contact metric manifolds and ( ) Sasakian manifolds, respectively [1,6,7]. Also, in 1989, Matsumoto [8] replaced the structure vector field ξ by - ξ in an almost paracontact manifold and associated a Lorentzian metric with the resulting structure and called it a Lorentzian almost paracontact manifold. In a Lorentzian almost paracontact manifold given by Matsumoto, the semi-riemannian metric has only index 1 and the structure vector field ξ is always lime like. Motivated by these circumstances M.M. Tripathi et. al. [9] associated a semi-riemannian metric, not necessarily Lorentzian, with an almost paracontact structure, and they called this indefinite almost paracontact metric structure an ( ) almost paracontact structure, where the structure vector field ξ is space like or time like according as = 1 or = 1. In [9] the authors para Sasakian studied ( ) almost para contact manifolds, and in particular ( ) manifold. They gave basic definitions and some examples of ( ) almost paracontact manifolds and introduced the notion of an ( ) para Sasakian structure. The basic properties, some typical, identities for curvature tensor and Ricci tensor of ( ) para Sasakian manifold were also studied in [9]. The authors proved that if a semi-riemannian manifold is one of flat proper recurrent or para Sasakian structure. proper Ricci-Recurrent than it can not admit an ( ) Also, they showed that, for an ( ) para Sasakian manifold, the conditions of being symmetric, semi-symmetric or of constant sectional curvature are all identical. In subsequent paper, M.M. Tripathi et.al studied 3 dimensional ( ) para Sasakian manifolds. They obtained a necessary and sufficient condition for an ( ) para Sasakian 3 manifolds to be an indefinite space form. The present paper is the continuation of previous studies. In section 2, we para sketch the notion of ( ) almost paracontact metric manifold, ( ) Sasakian manifold and its properties. In section-3, we introduce special ( ) para Sasakian manifold and prove that it is an ( ) para Sasakian manifold. Section-4, deals with recurrence properties of special ( ) para Sasakian manifold and finally in section-5, we study projective curvature tensor of special ( ) para Sasakian manifold. 2 Preliminaries Let M be an n-dimensional almost paracontact manifold [4] equipped with an almost paracontact structure ( ϕ, ξ, η) consisting of a tensor field of ϕ type (1, 1), a vector field ξ and a 1-form η satisfying 2 ϕ = I η ξ, (2.1) η ξ =, (2.2) ( ) 1

3 On indefinite almost paracontact metric manifold 1073 ( ξ ) = 0 φ, (2.3) and η 0 ϕ = 0. (2.4) Throughout this paper, we assume that, U, V, W χ( M), where χ ( M) is the Lie algebra of vector fields in M, unless specifically stated otherwise. By a semi-riemannian metric [2] on a manifold M, we understand a non-degenerate symmetric tensor field g of type (0, 2). In particular, if its index is 1, it becomes a Lorentzian metric [5]. Let g be a semi-riemannian metric with index (g) = v in an n-dimensional almost paracontact manifold M such that g( ϕ, ϕy ) = g(, Y ) ) Y ), (2.5) where = ± 1. Then M is called an ( ) almost paracontact metric manifold equipped with an ( ) almost paracontact metric structure ( ϕ, ξ, η, g, ε ) [9]. In particular, if index ( g ) = 1, then an ( ) almost paracontact metric manifold will be called a Lorentzian almost paracontact manifold. In particular, if the metric is positive definite, then an ( ) almost paracontact metric manifold is the usual almost paracontact metric manifold [4]. The condition (2.5) is equivalent to g(, ϕ Y ) = g( ϕ, Y ) (2.6) along with g(, ξ ) = ). (2.7) From (2.7), if follows that g ( ξ,ξ ) =. (2.8) that is, structure vector field ξ is never light like. Defining φ (, Y ) = g (, ϕ Y ), (2.9) we note that φ(, ξ ) = 0. (2.10) From (2.9), we also have φ)( Z ) = g( ϕ) ( Z ) = φ) ( Y ). (2.11) If on an ( ) almost paracontact metric manifold M 2 φ(, Y ) = η)( Y ) + Yη)( ) (2.12) for all, Y TM, then M is called as ( ) paracontact metric manifold [9]. An ( ) almost paracontact metric structure ( ϕ, ξ, η, g, ) is called an ( ) s paracontact metric structure if ξ = ϕ. (2.13) A manifold equipped with an ( ) S paracontact metric structure is called an ( ) S paracontact metric manifold. Equation (2.13) is equivalent to φ, Y = g ϕ, Y = g ξ, Y = η Y (2.14) ( ) ( ) ( ) ( )( )

4 1074 K. P. Pandey and B. Tiwari for all, Y TM. An ( ) almost paracontact metric structure is called an ( ) para Sasakian structure if [9] 2 ( ϕ )( Y ) = g ( ϕ, ϕy ) ξ η ( Y ) ϕ, (2.14) where is the Levi Civita connection with respect to g. A manifold endowed with an ( ) para Sasakian structure is called an ( ) para Sasakian manifold. For = 1 and g Riemannian, M is the usual para Sasakian manifold [4]. For, = 1, g Lorentzian and ξ replaced by - ξ, M becomes a Lorentzian Para Sasakian manifold [8]. para Sasakian manifold [9] R (, ξ ) = ) Y Y ), (2.16) R(, ξ ) = η ( ) g( + Y ) g(,, (2.17) R(, ) = ) g( + Y ) g(, (2.18) R ( ξ,, Y ) = g(, Y ) ξ + Y ), (2.19) and S(, ξ ) = ( n 1) ). (2.20) It is known that on an ( ) 3 Special ( ) Para Sasakian Manifold Consider an n-dimensional manifold admitting a 1 formη, a vector field ξ and a pseudo Riemannian metric g satisfying η )( Y ) = g(, Y ) + ) Y ) (3.1) and g(, ξ ) = ). (3.2) Further, if we put ξ = ϕ( ), (3.3) where ϕ is tensor field of type (1,1), then g( ϕ, Y ) = g( ξ, Y ). (3.4) The equation (3.2), yields η)( Y ) = g( ξ, Y ), (3.5) which, together with (3.4), implies φ(, Y ) = g( ϕ, Y ) = η)( Y ). (3.6) Form (3.6) and (3.1), we get = [ + η ( )ξ ], (3.7) which produces easily = η ( )ξ. (3.8) Also, (3.7) yields η ( ξ ) = 0 (3.9) and g(, Y ) = g(, Y ) ε ) Y ) (3.10)

5 On indefinite almost paracontact metric manifold 1075 Differentiating (3.7) covariantly, we obtain ϕ)( Y ) = ε[( η)( Y ) ξ + Y ) ξ ], which, in view of (3.3) and (3.1) reduces to 2 ϕ)( Y ) = g(, Y ) ξ ε Y ) ϕ. (3.11) All above results prove that the manifold satisfying (3.1), (3.2) and (3.3) is an ( ) para Sasakian manifold. We call this as special ( ) para Sasakian manifold. Theorem 3.1: On special ( ) para Sasakian manifold, the 1 form η - is closed. Proof: The proof of the theorem is obvious. Theorem3.2: On special ( ) para Sasakian manifold M, the vector field ξ is 2- killing vector field. Proof: A vector field is said to be 2-Killing if it satisfies [15] R(,, ξ, ) = g( ξ, ) + g( ξ, ξ ). (3.7) ξ ξ From (2.17), we get R ( ξ,, ξ, ) = g(, ). (3.8) Again, we have g ξ, ) + g( ξ, ξ ) g( ϕ, ϕ ). (3.9) ξ = From (3.8) and (3.9), we get R(,, ξ, ) = g( ξ, ) + g( ξ, ξ ), ξ ξ which implies that ξ is 2-killilng vector field. 4 Recurrent Special ( ) Para Sasakian Manifold In this section, we consider a special ( ) Para Sasakian Manifold, which satisfies R)( U ) = ) R( U ). (4.1) Theorem 4.1: If a special ( ) para Sasakian manifold satisfies (4.1) then. R(, Y, Z ) = [ g ( Z, Y ) η ( Y ) η ( Z )] + [ g ( Z, ) + η ( ) η ( Z )]Y. Proof: Let special ( ) para Sasakian manifold M satisfies R )( Y, Z, W ) = η ( ) R ( Y, Z, W ). Now, putting W = ξ in (4.1), we get R )( Y, Z, ξ ) = η ( ) R ( Y, Z, ξ ). (4.3) Using (2.16) in above, we get R )( Y, Z, ξ ) = η ( Y ) Z η ( Z ) Y η (. (4.4) [ ] ) Differentiating (2.16) covariantly, we get R)( ξ ) + R( ϕ ) = η)( Y ) Z η)( Z Y. (4.5) )

6 1076 K. P. Pandey and B. Tiwari Using (4.4) in (4.5), we get R( Y, Z, ϕ ) = η )( Y ) Z η )( Z ) Y η ( )( η ( Y ) Z η ( Z ) Y ), which in view of (3.1), gives R ( Y, Z, ϕ ) = g (, Y ) Z g (, Z ) Y (4.6) Using ( 3.7) in above, we get R(, Y, Z ) = [ g ( Y, Z ) η ( Y ) η ( Z )] + [ g (, Z ) + η ( ) η ( Z )]Y (4.7) Theorem 4.2: If special ( ) para Sasakian manifold satisfies (4.1) then it is η - Einstein manifold with constant associated scalar. Proof: An ( ) paracontact manifold is called η Einstein if its Ricci tensor satisfies S( = ag( b Y ) for some smooth functions a and b. Now, contracting (4.7), we get S( = ( n + 1) g( ( n 1) Y ). (4.8) This proves the result. Corollary 4.1: Special ( ) para Sasakian manifold satisfying (4.1) is a manifold of constant scalar curvature. Proof: From (4.8), we have r = n( n + 1) ε ( n 1). (4.9) 5 Projective Curvature Tensor Projective curvature tensor of special ( ) para Sasakian manifold is given by 1, = R(, [ S( S(, Y ], (5.1) ( n 1) where Q is Ricci map is given by S ( = g( Q, Y ). Analogous to [10], we define an ( ) almost para contact manifold M is said to be Quasi projectively flat if g(,, ϕ W ) = 0 (5.2) and ξ -projectively flat if P (, ξ ) = 0. (5.3) Theorem 5.1: Special ( ) Sasakian para contact manifold M is ξ projectively flat. Proof: Putting Z = ξ in above we get 1 P (, ξ ) =, ξ ) ξ ) n 1 Now using (2.16) and (2.20) is above, we get [ S( ξ ) S(, Y ]

7 On indefinite almost paracontact metric manifold , ξ ) = ) Y Y ) η ( n 1) which gives P (, ξ ) = 0. Theorem 5.2: On special ( ) Sasakian paracontact manifold M, we have P (, ξ ) = 0, P (, W ) = g,, W. [ ( n 1) Y ) + ( n 1) ( ) Y ] where ( ) Proof: Using (2.7), (2.18) and (2.20), we obtain, ξ ) = R(, ) [ g( ) + g(, Y )] which implies P (, ξ ) = 0. (5.4) Theorem 5.3: Special ( ) para Sasakian manifold is quasi-protectively flat if and only if it is projectively flat.. Proof: In view of equation (3.7), we obtain g (,, ϕ ) = [ g(,, W ) + W ) g(,, ξ ], which due to (5.4), reduces to g(,, ϕ ) = [ g(,, W )]. From this, we see that the manifold is quasi-projectively flat if and only if it is projectively flat. Corollary 5.1: A quasi-projectively flat special ( ) para Sasakian manifold is Einstein. Proof: It is known [11] that a projectively flat manifold is an Einstein manifold. Now, from theorem (5.4), corollary follows. Theorem 5.4: If special ( ) para Sasakian manifold satisfies any two of the following three conditions then it satisfies the third one. (1) Manifold is quasi projectively flat, (2) g( ϕ,, ϕw ) = 0, (3) ξ,, Y ) = 0, then it satisfies the third one. Proof: Using equation (3.7), we get g( ϕ,, ϕw ) = g(,, ϕw ) + ) g( ξ,, ϕw ), In view of theorem (5.3), we see that if manifold satisfies any two of the above three conditions then it satisfies the third one. References 1. A. Bejancu and K. L. Duggal, Real hypersurfaces of indefinite Kaehler manifolds, International Journal of Mathematics and Mathematical Sciences, 16(3), , 1993.

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