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    a  r   X   i  v  :   1   6   1   2 .   0   3   2   9   7  v   1   [  m  a   t   h .   D   G   ]   1   0   D  e  c   2   0   1   6 ON WARPED PRODUCT MANIFOLDS SATISFYING SOMEPSEUDOSYMMETRIC TYPE CONDITIONS ABSOS ALI SHAIKH AND HARADHAN KUNDU Abstract.  The object of the present paper is to study the characterization of warped productmanifolds satisfying some pseudosymmetric type conditions, especially, due to projective curvaturetensor. For this purpose we consider a warped product manifold satisfying the pseudosymmetrictype condition  R · R  =  L 1 Q ( g,R ) + L 2 Q ( S,R ) and evaluate its characterization theorem. As spe-cial cases of   L 1  and  L 2  we find out the necessary and sufficient condition for a warped productmanifold to satisfy various pseudosymmetric type, such as pseudosymmetry, Ricci generalized pseu-dosymmetry, semisymmetry due to projective curvature tensor ( P   · R  = 0), pseudosymmetry dueto projective curvature tensor ( P   · R  =  LQ ( g,R )) etc. Finally we present some suitable examplesof warped product manifolds satisfying such pseudosymmetric type conditions. 1.  Introduction Let  ∇ ,  R ,  S  ,  G ,  P   and  κ  be respectively the Levi-Civita connection, the Riemann-Christoffelcurvature tensor, Ricci tensor, the Gaussian curvature tensor, the projective curvature tensor andthe scalar curvature of an  n -dimensional ( n  ≥  3) connected smooth semi-Riemanian manifold  M  equipped with the semi-Riemannian metric  g . Symmetry is a very important geometric property of a space. Cartan [3] introduced the notion of symmetry (local and global) on a Riemannian manifoldin terms of geodesic symmetries. A semi-Riemannian manifold is said to be locally symmetric [3] if its local geodesic symmetries at each point are all isometry. According to Cartan-Ambrose-Hickstheorem a locally symmetric manifold can be characterized by the curvature condition ∇ R  = 0, i.e.,the curvature tensor is covariantly constant (For the meaning and definition of various notationsand symbols used here, we refer the reader to Section 2 of this paper).As a proper generalization of locally symmetric manifold, Cartan [4] introduced the notion of semisymmetric manifold. A semi-Riemannian manifold is said to be semisymmetric [4] (see also[30], [31], [32]) if   R · R  = 0, where the first  R  stands for the curvature operator acting as a derivationon the second  R . It may be noted that a semi-Riemannian manifold is semisymmetric if and onlyif its sectional curvature function  k (  p,π ) is invariant up to second order, under parallel transportof any plane  π  at any point  p  of   M   around any infinitesimal coordinate parallelogram centered at  p  (see [12], [15], [16]). Date  : December 13, 2016.2010  Mathematics Subject Classification.  53C15, 53C25, 53C35. Key words and phrases.  Semisymmetric manifold, pseudosymmetric manifold, pseudosymmetric type manifold,warped product manifold. 1  2 A. A. SHAIKH AND H. KUNDU During the study of totally umbilical submanifolds of semisymmetric manifolds as well as duringthe consideration of geodesic mappings on semisymmetric manifolds, Adamow and Deszcz [1](see [7] and also references therein) introduced the notion of pseudosymmetric manifolds as aproper generalization of semisymmetric manifolds. A semi-Riemannian manifold is said to bepseudosymmetric if  R · R  and  Q ( g,R ) are linearly dependent.Let  M   be not of constant curvature and  U   denotes the set  { x  ∈  M   :  Q ( g,R )   = 0 } . Then at apoint  p  ∈  U  , a plane  π 1 (  v  p ,  w  p )  ⊂  T   p M   is said to be curvature dependent with respect to anotherplane  π 2 (  x  p ,  y  p )  ⊂  T   p M   if   Q ( g,R )(  v  p ,  w  p ,  v  p ,  w  p ,  x  p ,  y  p )   = 0. Now if   π 1  is curvature dependent withrespect to  π 2 , then the scalar L (  p,π 1 ,π 2 ) =  R · R (  v  p ,  w  p ,  v  p ,  w  p ,  x  p ,  y  p ) Q ( g,R )(  v  p ,  w  p ,  v  p ,  w  p ,  x  p ,  y  p )is called the double sectional curvature or Deszcz sectional curvature ([12], [15], [16], [17]) of the plane  π 1  with respect to  π 2  at  p . In terms of Deszcz sectional curvature, a semi-Riemannian man-ifold is pseudosymmetric if at each point  p  ∈  U  ,  L (  p,π 1 ,π 2 ) is independent of the planes  π 1  and π 2  (see [12], [15], [16], [17]). Replacing  R  by other curvature tensors and  g  by other symmetric (0 , 2)-tensor in the definingcondition of semisymmetric manifold and pseudosymmetric manifold one can get various curvaturerestricted geometric structures, which are simply called as semisymmetric type and pseudosym-metric type manifolds ([10], [11], [13], [22], [25], [28]). For the geometric meaning of Weyl semisym- metric and Weyl pseudosymmetric spaces we refer the reader to see [18]. One of the importantsemisymmetric type (resp., pseudosymmetric type) manifold is semisymmetric (resp., pseudosym-metric) manifold due to projective curvature tensor. A semi-Riemannian manifold is said to besemisymmetric (resp., pseudosymmetric) due to projective curvature tensor if  P   · R  = 0 (resp.,  P   · R  and  Q ( g,R ) are linearly dependent).Another important pseudosymmetric type manifold is Ricci generalized pseudosymmetric manifold.A semi-Riemannian manifold is said to be Ricci generalized pseudosymmetric ([5], [6]) if  R · R  and  Q ( S,R ) are linearly dependent.We refer the reader to see [26] for details about various curvature restricted geometric structuresdue to projective curvature tensor.Again the notion of warped product manifold ([2], [19]) is a generalization of product manifold and this notion is important due to its applications in general theory of relativity and cosmology.Various spacetimes are warped product, e.g., Robertson-Walker spacetimes, asymptotically flatspacetimes, Schwarzschild spacetimes, Kruskal space-times, Reissner-Nordstr¨om spacetimes etc.  ON WARPED PRODUCT PSEUDOSYMMETRIC TYPE MANIFOLDS 3 The main purpose of this present paper is to study the characterization of a warped product semi-Riemannian manifold realizing some pseudosymmetric type curvature conditions, especially, due tothe projective curvature tensor. For this purpose we consider the pseudosymmetric type condition R · R  =  L 1 Q ( g,R ) +  L 2 Q ( S,R ), and evaluate the characterization of a warped product manifoldsatisfying such curvature condition. As a special case we get the characterization of a warpedproduct manifold which is (i) semisymmetric, (ii) pseudosymmetric, (iii) Ricci generalized pseu-dosymmetric, (iv) special Ricci generalized pseudosymmetric, (v) semisymmetric due to projectivecurvature tensor and (vi) pseudosymmetric due to projective curvature tensor etc. It is shown thatif a warped product manifold  M   =  M   × f    M   satisfies  R  · R  =  L 1 Q ( g,R ) +  L 2 Q ( S,R ) such that L 2  is nowhere zero, then at either the base  M   is flat or the fiber   M   is Einstein. Consequently fora warped product pseudosymmetric manifold due to projective curvature tensor or special Riccigeneralized pseudosymmetric manifold either the base is flat or the fiber is Einstein.The paper is organized as follows. After discussing various notations as preliminaries in Section2, we define various pseudosymmetric type curvature restricted geometric structures in Section3. Section 4 is devoted to the study of warped product manifold and we state the curvature re-lation of a warped product manifold with its base and fiber. In Section 5 we discuss about thecharacterization theorems of various pseudosymmetric type warped product manifolds. Finally tosupport our results we present some suitable examples of warped product manifolds in the lastsection. It is interesting to mention that notion of pseudosymmetric manifold arose during thestudy of totally umbilical hypersurface of a semisymmetric manifold, and in Example 1 we presenta pseudosymmetric totally umbilical hypersurface of a semisymmetric manifold.2.  Preliminaries Let  M   be a connected  n -dimensional smooth manifold equipped with the semi-Riemannianmetric  g . Let us consider the following notations related to ( M,g ): C  ∞ ( M  ) = the algebra of all smooth functions on  M  , T   rk  ( M  ) = the space of all smooth tensor fields of type ( r,k ) on  M   and χ ( M  ) =  T   10  ( M  ) = the Lie algebra of all smooth vector fields on  M  .The Kulkarni-Nomizu product ([9], [14], [29])  A ∧ E   ∈ T   04  ( M  ) of   A  and  E   ∈ T   02  ( M  ), is given by( A ∧ E  )( X  1 ,X  2 ,X  3 ,X  4 ) =  A ( X  1 ,X  4 ) E  ( X  2 ,X  3 ) +  A ( X  2 ,X  3 ) E  ( X  1 ,X  4 ) −  A ( X  1 ,X  3 ) E  ( X  2 ,X  4 ) − A ( X  2 ,X  4 ) E  ( X  1 ,X  3 ) , where  X  1 ,X  2 ,X  3 ,X  4  ∈  χ ( M  ). Throughout the paper we consider  X,Y,X  1 ,X  2 , ···∈  χ ( M  ).Now for  D  ∈ T   04  ( M  ),  A  ∈ T   02  ( M  ) and  X,Y   ∈  χ ( M  ), we get two endomorphisms  D  ( X,Y  )(called the associated curvature operator of   D ) and  X   ∧ A  Y   defined by D  ( X,Y  )( X  1 ) =  D ( X,Y  ) X  1  and  4 A. A. SHAIKH AND H. KUNDU ( X   ∧ A  Y   ) X  1  =  A ( Y,X  1 ) X   − A ( X,X  1 ) Y, where  D ∈ T   13  ( M  ) such that  g ( D ( X,Y  ) X  1 ,X  2 ) =  D ( X,Y,X  1 ,X  2 ), called the associated (1 , 3)tensor of   D .A tensor  D  ∈ T   04  ( M  ) is said to be a generalized curvature tensor ([9], [20], [22]) if  D ( X  1 ,X  2 ,X  3 ,X  4 ) +  D ( X  2 ,X  3 ,X  1 ,X  4 ) +  D ( X  3 ,X  1 ,X  2 ,X  4 ) = 0 ,D ( X  1 ,X  2 ,X  3 ,X  4 ) +  D ( X  2 ,X  1 ,X  3 ,X  4 ) = 0 and D ( X  1 ,X  2 ,X  3 ,X  4 ) =  D ( X  3 ,X  4 ,X  1 ,X  2 ) . The Gaussian curvature tensor  G , Weyl conformal curvature tensor  C  , concircular curvature tensor W   and conharmonic curvature tensor  K  are all generalized curvature tensors and respectively givenby G  = 12 g ∧ g,C   =  R −  1 n − 2 g ∧ S   +  κ 2( n − 1)( n − 2) g ∧ g,W   =  R −  κ 2 n ( n − 1) g ∧ g  and K   =  R −  1 n − 2 g ∧ S. The projective curvature tensor  P   of type (0 , 4) , given by P  ( X  1 ,X  2 ,X  3 ,X  4 ) =  R ( X  1 ,X  2 ,X  3 ,X  4 ) −  1 n − 2 [ S  ( X  2 ,X  3 ) g ( X  1 ,X  4 ) − S  ( X  1 ,X  3 ) g ( X  2 ,X  4 )] , is not a generalized curvature tensor.Again an endomorphism  L    can be operate on a (0 ,k )-tensor  H   and obtain  L   H   as follows:( L   H  )( X  1 ,X  2 , ···  ,X  k ) =  − H  ( L   X  1 ,X  2 , ···  ,X  k ) −···− H  ( X  1 ,X  2 , ···  , L   X  k ) . In particular, for  L    =  D  ( X,Y   ) and  X   ∧ A  Y   we get two (0 ,k  + 2) tensors  D  · H   and  Q ( A,H  )defined as ([23], [28], [11] and also references therein) D · H  ( X  1 ,X  2 ,...,X  k ,X,Y  ) = ( D  ( X,Y  ) · H  )( X  1 ,X  2 ,...,X  k )=  − H  ( D ( X,Y  ) X  1 ,X  2 ,...,X  k ) −···− H  ( X  1 ,X  2 ,..., D ( X,Y  ) X  k ) ,Q ( A,H  )( X  1 ,X  2 , ···  ,X  k ,X,Y   ) = (( X   ∧ A  Y  ) · H  )( X  1 ,X  2 ,...,X  k )=  A ( X,X  1 ) H  ( Y,X  2 , ···  ,X  k ) + ··· +  A ( X,X  k ) H  ( X  1 ,X  2 , ···  ,Y  ) − A ( Y,X  1 ) H  ( X,X  2 , ···  ,X  k ) −···− A ( Y,X  k ) H  ( X  1 ,X  2 , ···  ,X  ) . Lemma 2.1.  Let   M   be a connected warped product manifold with base   M   and fiber    M  . If   f  1  ∈ C  ∞ ( M  )  and   f  2  ∈  C  ∞ (  M  )  satisfies   f  1  f  2  ≡  0 , then either   f  1  ≡  0  or   f  2  ≡  0 .
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