On warped product manifolds satisfying some pseudosymmetric type conditions
aa r X i v : . [ m a t h . DG ] D ec 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 Q ( g, R ) + L Q ( S, R ) and evaluate its characterization theorem. As spe-cial cases of L and L 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. 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 ofa 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] ifits 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 ofsemisymmetric 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 : September 24, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Semisymmetric manifold, pseudosymmetric manifold, pseudosymmetric type manifold,warped product manifold.
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 ( ~v p , ~w p ) ⊂ T p M is said to be curvature dependent with respect to anotherplane π ( ~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 is curvature dependent withrespect to π , then the scalar L ( p, π , π ) = 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 theplane π with respect to π at p . In terms of Deszcz sectional curvature, a semi-Riemannian man-ifold is pseudosymmetric if at each point p ∈ U , L ( p, π , π ) is independent of the planes π and π (see [12], [15], [16], [17]).Replacing R by other curvature tensors and g by other symmetric (0 , 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 manifoldand 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.
N 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 Q ( g, R ) + L 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 f M satisfies R · R = L Q ( g, R ) + L Q ( S, R ) such that L is nowhere zero, then at either the base M is flat or the fiber f 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 ( M ) = the Lie algebra of all smooth vector fields on M .The Kulkarni-Nomizu product ([9], [14], [29]) A ∧ E ∈ T ( M ) of A and E ∈ T ( M ), is given by( A ∧ E )( X , X , X , X ) = A ( X , X ) E ( X , X ) + A ( X , X ) E ( X , X ) − A ( X , X ) E ( X , X ) − A ( X , X ) E ( X , X ) , where X , X , X , X ∈ χ ( M ). Throughout the paper we consider X, Y, X , X , · · · ∈ χ ( M ).Now for D ∈ T ( M ), A ∈ T ( 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 ) = D ( X, Y ) X and A. A. SHAIKH AND H. KUNDU ( X ∧ A Y ) X = A ( Y, X ) X − A ( X, X ) Y, where D ∈ T ( M ) such that g ( D ( X, Y ) X , X ) = D ( X, Y, X , X ), called the associated (1 , D .A tensor D ∈ T ( M ) is said to be a generalized curvature tensor ([9], [20], [22]) if D ( X , X , X , X ) + D ( X , X , X , X ) + D ( X , X , X , X ) = 0 ,D ( X , X , X , X ) + D ( X , X , X , X ) = 0 and D ( X , X , X , X ) = D ( X , X , X , X ) . 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 − n − g ∧ S + κ n − n − g ∧ g,W = R − κ n ( n − g ∧ g and K = R − n − g ∧ S. The projective curvature tensor P of type (0 ,
4) , given by P ( X , X , X , X ) = R ( X , X , X , X ) − n − S ( X , X ) g ( X , X ) − S ( X , X ) g ( X , X )] , 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 , X , · · · , X k ) = − H ( L X , X , · · · , X k ) − · · · − H ( X , X , · · · , 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 , X , . . . , X k , X, Y ) = ( D ( X, Y ) · H )( X , X , . . . , X k )= − H ( D ( X, Y ) X , X , . . . , X k ) − · · · − H ( X , X , . . . , D ( X, Y ) X k ) ,Q ( A, H )( X , X , · · · , X k , X, Y ) = (( X ∧ A Y ) · H )( X , X , . . . , X k )= A ( X, X ) H ( Y, X , · · · , X k ) + · · · + A ( X, X k ) H ( X , X , · · · , Y ) − A ( Y, X ) H ( X, X , · · · , X k ) − · · · − A ( Y, X k ) H ( X , X , · · · , X ) . Lemma 2.1.
Let M be a connected warped product manifold with base M and fiber f M . If f ∈ C ∞ ( M ) and f ∈ C ∞ ( f M ) satisfies f f ≡ , then either f ≡ or f ≡ . N WARPED PRODUCT PSEUDOSYMMETRIC TYPE MANIFOLDS 5
Lemma 2.2. [5]
Let
A, E ∈ T ( M ) be symmetric. Then Q ( A, E ) = 0 if and only if A and E arelinearly dependent [5] . Moreover Q ( A, E ) = 0 if and only if M is Einstein, i.e., S = κn g . Pseudosymmetric type curvature restricted geometric structures
Definition 3.1. ( [22] , [30] , [31] , [32] ) For H ∈ T k ( M ) and D ∈ T ( M ) , a semi-Riemannianmanifold M is said to be H -semisymmetric type if D · H = 0 . In particular, a semi-Riemannian manifold with the semisymmetric type conditions R · R = 0, R · S = 0, R · P = 0, P · R = 0 and P · S = 0 is respectively called semisymmetric, Riccisemisymmetric, projective semisymmetric, semisymmetric due to projective curvature tensor andRicci semisymmetric due to projective curvature tensor respectively. Definition 3.2. ([1] , [7] , [9] , [22]) For H ∈ T k ( M ) and D i ∈ T ( M ) , i = 1 , , · · · r, r ≥ , a semi-Riemannian manifold is said to be H -pseudosymmetric type if D i · H are linearly dependent, i.e., r P i =1 c i ( D i · H ) = 0 for some c i ∈ C ∞ ( M ) , called the associated scalars. Moreover if the associatedscalars are constant then the manifold is called pseudosymmetric type manifold of constant type. In particular, a semi-Riemannian manifold satisfying R · C = L R Q ( g, C ) on { x ∈ M : C x = 0 } ,R · S = L S Q ( g, S ) on n x ∈ M : (cid:16) S − κn g (cid:17) x = 0 o ,R · P = L P Q ( g, P ) on { x ∈ M : Q ( g, P ) x = 0 } ,P · R = L Q ( g, R ) on (cid:26) x ∈ M : (cid:18) R − κn ( n − G (cid:19) x = 0 (cid:27) and P · S = L Q ( g, S ) on n x ∈ M : (cid:16) S − κn g (cid:17) x = 0 o is called conformally pseudosymmetric, Ricci pseudosymmetric, projective pseudosymmetric, pseu-dosymmetric due to projective curvature tensor and Ricci pseudosymmetric due to projective cur-vature tensor respectively, where L R , L S , L P , L , L are the associated scalars.In this paper we are mainly interested on the pseudosymmetric type condition R · R = L Q ( g, R )+ L Q ( S, R ). For particular values of L and L , we get various pseudosymmetric type conditionsas follows: A. A. SHAIKH AND H. KUNDU
Values of L & L Structure name Curvature condition L = L = 0 Semisymmetric manifold R · R = 0 L = 0 Pseudosymmetric manifold R · R = L Q ( g, R ) L = 0 Ricci generalizedPseudosymmetric manifold R · R = L Q ( S, R ) L = 0 , L = 1 Special Ricci generalizedPseudosymmetric manifold R · R = Q ( S, R ) L = κn ( n − , L = 0 Semisymmetric due toconcircular curvature tensor W · R = 0 L = κn ( n − Ricci generalized Pseudosymmetricdue to concircular curvature tensor W · R = L Q ( S, R ) L = 0 , L = n − Semisymmetric due toprojective curvature tensor P · R = 0 L = n − Pseudosymmetric due toprojective curvature tensor P · R = L Q ( g, R )4. Warped product manifolds
Warped product is an important notion in semi-Riemannian geometry and it has a great ap-plications in general theory of relativity and cosmology. To impose a semi-Riemannian structureon a product smooth manifold, the notion of warped product metric arose as a generalizationof Riemannian product metric. This notion was independently introduced by Kru˘ckovi˘c [19] (assemi-decomposable metric) as well as Bishop and O’Neill [2]. The most important example ofnon-Riemann product but a warped product is surface of revolution. Many well-known space-times, e.g., Schwarzschild, Kottler, Reissner-Nordstr¨om, Reissner-Nordstr¨om-de Sitter, Vaidya,Robertson-Walker spacetimes are all warped products.Let (
M , g ) and ( f M , e g ) be two semi-Riemannian manifolds of dimension p and ( n − p ) respectively(1 ≤ p ≤ n −
1) and M = M × f M . The warped product metric g on M is given by g = π ∗ ( g ) + ( f ◦ π ) σ ∗ ( e g ) , where f is a positive smooth function on M and π : M → M (resp., σ : M → f M ) is the naturalprojection on M (resp., f M ). The manifold M is called the base, f M is called the fiber and f iscalled the warping function of M . If f = 1, then the warped product reduces to the Riemannproduct. If we consider a product chart ( U × V ; x , x , ..., x p , x p +1 = y , x p +2 = y , ..., x n = y n − p ) N WARPED PRODUCT PSEUDOSYMMETRIC TYPE MANIFOLDS 7 on M , then in terms of local coordinates, g can be expressed as g ij = g ij for i = a and j = b,f e g ij for i = α and j = β ,0 otherwise,(4.1)where a, b ∈ { , , ..., p } and α, β ∈ { p + 1 , p + 2 , ..., n } . We note that throughout the paper weconsider a, b, c, d, e, s, t ∈ { , , ..., p } ; α, β, γ, δ, ǫ, µ, η ∈ { p + 1 , p + 2 , ..., n } and i, j, k, l, q, u, v ∈{ , , ..., n } . Moreover, when (cid:3) is a quantity formed with respect to g , we denote by (cid:3) and e (cid:3) , thesimilar quantities formed with respect to g and e g respectively.By a straightforward calculation we can evaluate the components of various necessary tensorsof a warped product manifold in terms of the base and fiber components. The non-zero localcomponents R hijk of the Riemann-Christoffel curvature tensor R , S jk of the Ricci tensor S and thescalar curvature κ of M are respectively given by R abcd = R abcd , R aαbβ = f T ab e g αβ , R αβγδ = f e R αβγδ − f ∆ e G αβγδ ,S ab = S ab − ( n − p ) T ab , S αβ = e S αβ + Ω e g αβ and κ = κ + e κf − ( n − p )[( n − p − − tr ( T )] , where G ijkl = g il g jk − g ik g jl are the components of Gaussian curvature and T ab = 12 f ( f a,b − f f a f b ) , tr ( T ) = g ab T ab , ∆ = 14 f g ab f a f b , Ω = − f (( n − p − tr ( T )) , f a = ∂ a f = ∂f∂x a . The non-identically zero local components of R · R , Q ( g, R ) and Q ( S, R ) of M are given by( R · R ) abcdst = ( R · R ) abcdst , (4.2) ( R · R ) aαbβst = f e g αβ ( R · T ) abst , ( R · R ) abcαsη = f e g αη (cid:0) T as T bc − T ac T bs + T ts R abct (cid:1) , ( R · R ) aαβγsη = − f T as [ e R ηαβγ − f ∆ e G ηαβγ ] − f T as e G ηαβγ , ( R · R ) αβγδµη = f [( e R · e R ) αβγδµη − f ∆ Q ( e g, e R ) αβγδµη ]; A. A. SHAIKH AND H. KUNDU Q ( g, R ) abcdst = Q ( g, R ) abcdst , (4.3) Q ( g, R ) aαbβst = f e g αβ Q ( g, T ) abst ,Q ( g, R ) abcαsη = f e g αη (cid:0) R abct − g bs T ac + g as T bc (cid:1) ,Q ( g, R ) aαβγsη = − f [ g as e R ηαβγ + f ( T as − ∆ g as ) e G ηαβγ ] ,Q ( g, R ) αβγδµη = f Q ( e g, e R ) αβγδµη ; Q ( S, R ) abcdst = Q ( S, R ) abcdst − ( n − p ) Q ( S, R ) abcdst , (4.4) Q ( S, R ) aαbβst = f e g αβ Q ( S, T ) abst ,Q ( S, R ) abcαsη = R abcs [ e S αβ + Ω e g αβ ]+ f e g αη [ T bc ( S as − ( n − p ) T as ) − T ac ( S bs − ( n − p ) T bs )] ,Q ( S, R ) aαβγsη = − f ( S as − ( n − p ) T as )( e R ηαβγ − f ∆ e G ηαβγ ) − f T as he g αβ ( e S γη + Ω e g γη ) − e g αγ ( e S βη + Ω e g βη ) i ,Q ( S, R ) aαbβµη = − f T ab Q ( e g, e S ) αβµη ,Q ( S, R ) αβγδµη = f [ Q ( e S, e R ) αβγδµη − ∆ Q ( e S, e G ) αβγδµη + f Ω Q ( e g, e R ) αβγδµη ] . We refer the readers to see [5], [21], [24], [27] and also references therein for detail informationabout warped product components of various tensors on M .5. Main results
Theorem 5.1.
Let M n = M p × f f M n − p be a warped product manifold. Then M satisfies thepseudosymmetric type condition (5.1) R · R = L Q ( g, R ) + L Q ( S, R ) if and only if the following conditions hold simultaneously: ( I ) R · R = L Q ( g, R ) + L Q ( S, R ) − L ( n − p ) Q ( T, R ) , ( II ) f e g αβ (cid:0) T as T bc − T ac T bs + T ts R abct (cid:1) = L f e g αβ (cid:0) R abcs − T ac g bs + T bc g as (cid:1) + L R abcs (cid:16) e S αβ + Ω e g αβ (cid:17) + L f e g αβ (cid:2) T bc (cid:0) S as − ( n − p ) T as (cid:1) − T ac (cid:0) S bs − ( n − p ) T bs (cid:1)(cid:3) , ( III ) T as (cid:16) e R ηαβγ − f ∆ e G ηαβγ (cid:17) + f T as e G ηαβγ = L h g as e R ηαβγ + f ( T as − ∆ g as ) e G ηαβγ i + L ( S as − ( n − p ) T as )( e R ηαβγ − f ∆ e G ηαβγ )+ L T as he g αβ ( e S γη + Ω e g γη ) − e g αγ ( e S βη + Ω e g βη ) i , ( IV ) L T ab Q ( e g, e S ) αβγδ = 0 and ( V ) e R · e R = f (∆ + L + Ω L ) Q ( e g, e R ) + L Q ( e S, e R ) − L ∆ Q ( e S, e G ) . N WARPED PRODUCT PSEUDOSYMMETRIC TYPE MANIFOLDS 9
Proof:
In terms of local coordinates, (5.1) can be written as(5.2) ( R · R ) ijkluv = L Q ( g, R ) ijkluv + L Q ( S, R ) ijkluv . Now from (4.2), (4.3) and (4.4) we see that the non-zero possibilities of (5.2) are(i) i = a, j = b, k = c, l = d, u = s, v = t ,(ii) i = a, j = α, k = b, l = β, u = s, v = t ,(iii) i = a, j = b, k = c, l = α, u = s, v = η ,(iv) i = a, j = α, k = β, l = γ, u = s, v = η ,(v) i = a, j = α, k = b, l = β, u = µ, v = η and(vi) i = α, j = β, k = γ, l = δ, u = µ, v = η .So it is obvious that to prove the theorem we have only to show that the conditions (I)-(V) arethe simplified form of the possibilities (i)-(vi). Putting (i), (iii), (iv), (v) and (vi) in (5.2) andsimplifying we get (I) to (V) respectively. Again putting (ii) in (5.2), we get R · T = L Q ( g, T ) + L Q ( S, T ), which obviously follows from (II). This completes the proof.
Corollary 5.1.
If a warped product manifold M n = M p × f f M n − p satisfies the pseudosymmetrictype condition R · R = L Q ( g, R ) + L Q ( S, R ) , then R · T = L Q ( g, T ) + L Q ( S, T ) . Proof:
From condition (II) of Theorem 5.1 we get the result easily.
Theorem 5.2. If M n = M p × f f M n − p is a warped product manifold satisfying (5.1) , then M = { x ∈ M : R | π ( x ) = 0 } ∪ { x ∈ M : ( T − L g ) | π ( x ) = 0 } ∪ { x ∈ M : ( e S − e κn − p e g ) | σ ( x ) = 0 } . Proof:
Since M satisfies (5.1), then from the condition (IV) of Theorem 5.1, at every point x ∈ M we have the following three cases:Case 1: L = 0. Therefore from (III) of Theorem 5.1, we get( T − L g ) e R = f (cid:2) L ( T − ∆ g ) + ∆ T − T (cid:3) e G ⇒ either T − L g = 0 or e R is a scalar multiple of e G and thus e S = e κn − p e g at x .Case 2: T = 0. In this case putting the value of T in (II) of Theorem 5.1, we get R ( L f e g + L e S + L Ω e g ) = 0 ⇒ either R = 0 or L e S + ( L f + L Ω) e g = 0 at x ⇒ either R = 0 or L = 0 or e S = e κn − p e g at x. Case 3: Q ( e g, e S ) = 0. Then from Lemma 2.1 of [5], e S = e κn − p e g at x .Now R = 0 at x means R | π ( x ) = 0, T − L g = 0 at x means ( T − L g ) | π ( x ) = 0 and e S − e κn − p e g = 0at x means ( e S − e κn − p e g ) | σ ( x ) = 0. Now combining the resulting condition of above cases we get ourassertion. Theorem 5.3.
If a warped product manifold M n = M p × f f M n − p satisfies the pseudosymmetrictype condition (5.1) and L is nowhere zero, theneither (i) the base M is flator (ii) the fiber f M is Einstein. Proof:
Since M satisfies (5.1) and L is nowhere zero, then from the condition (IV) of Theorem5.1, T Q ( e g, e S ) ≡ ⇒ T ≡ Q ( e g, e S ) ≡ T is identically zero, then from Condition (II), R ( L f e g + L e S + L Ω e g ) ≡ ⇒ R ≡ L f e g + L e S + L Ω e g ≡ . [by Lemma 2.1] ⇒ base is flat or fiber is Einstein [as L is nowhere zero].Again if Q ( e g, e S ) ≡ f M is Einstein. This completes the proof.Now from Theorem 5.1 we can easily get the characterization for a warped product semisym-metric and various pseudosymmetric type manifolds as follows: Corollary 5.2.
Let M n = M p × f f M n − p be a warped product manifold. Then M satisfies R · R = 0 if and only if the following conditions hold simultaneously:(I) R · R = 0 , (II) T as T bc − T ac T bs + f T ts R abct = 0 , (III) T as (cid:16) e R ηαβγ − f ∆ e G ηαβγ (cid:17) + T as e G ηαβγ = 0 , (IV) e R · e R = f ∆ Q ( e g, e R ) . Corollary 5.3.
Let M n = M p × f f M n − p be a warped product semisymmetric manifold. Then(i) the base M is semisymmetric,(ii) the fiber f M is pseudosymmetric,(iii) the fiber f M is of constant curvature if T = 0 .(iv) R · T = 0 . Corollary 5.4. ( [6] , [8] , [11] ) Let M n = M p × f f M n − p be a warped product manifold. Then M satisfies R · R = L R Q ( g, R ) if and only if the following conditions hold simultaneously:(I) R · R = L R Q ( g, R ) , (II) T ts (cid:0) R abct − T ac g bt + T bc g at (cid:1) = L R (cid:0) R abcs − T ac g bs + T bc g as (cid:1) , (III) T as (cid:16) e R ηαβγ − f ∆ e G ηαβγ (cid:17) + f T as e G ηαβγ = L R h g as e R ηαβγ + f ( T as − ∆ g as ) e G ηαβγ i ,(IV) e R · e R = ( f ∆ + f L R ) Q ( e g, e R ) . Corollary 5.5.
Let M n = M p × f f M n − p be a warped product pseudosymmetric manifold ( R · R = L R Q ( g, R )) . Then N WARPED PRODUCT PSEUDOSYMMETRIC TYPE MANIFOLDS 11 (i) the base and fiber both are pseudosymmetric,(ii) R · T = L R Q ( g, T ) . Corollary 5.6.
Let M n = M p × f f M n − p be a warped product manifold. Then M satisfies R · R = LQ ( S, R ) if and only if the following conditions hold simultaneously:(I) R · R = LQ ( S, R ) − L ( n − p ) Q ( T, R ) , (II) f e g αβ (cid:0) T as T bc − T ac T bs + T ts R abct (cid:1) = LR abcs (cid:16) e S αβ + Ω e g αβ (cid:17) + Lf e g αβ (cid:2) T bc (cid:0) S as − ( n − p ) T as (cid:1) − T ac (cid:0) S bs − ( n − p ) T bs (cid:1)(cid:3) ,(III) T as (cid:16) e R ηαβγ − f ∆ e G ηαβγ (cid:17) + f T as e G ηαβγ = L ( S as − ( n − p ) T as )( e R ηαβγ − f ∆ e G ηαβγ )+ LT as he g αβ ( e S γη + Ω e g γη ) − e g αγ ( e S βη + Ω e g βη ) i ,(IV) LT ab Q ( e g, e S ) αβγδ = 0 and(V) e R · e R = ( f ∆ + Ω L ) Q ( e g, e R ) + LQ ( e S, e R ) − L ∆ Q ( e S, e G ) . Corollary 5.7.
If a warped product manifold M n = M p × f f M n − p satisfies R · R = LQ ( S, R ) , then M = { x ∈ M : R | π ( x ) = 0 } ∪ { x ∈ M : T | π ( x ) = 0 } ∪ { x ∈ M : ( e S − e κn − p e g ) | σ ( x ) = 0 } . Corollary 5.8. [6]
Let M n = M p × f f M n − p be a warped product manifold. Then M satisfies thespecial Ricci generalized pseudosymmetric condition R · R = Q ( S, R ) if and only if the followingconditions hold simultaneously:(I) R · R = Q ( S − ( n − p ) T, R ) , (II) f e g αβ (cid:0) T as T bc − T ac T bs + T ts R abct (cid:1) = R abcs (cid:16) e S αβ + Ω e g αβ (cid:17) + f e g αβ (cid:2) T bc (cid:0) S as − ( n − p ) T as (cid:1) − T ac (cid:0) S bs − ( n − p ) T bs (cid:1)(cid:3) ,(III) T as (cid:16) e R ηαβγ − f ∆ e G ηαβγ (cid:17) + f T as e G ηαβγ = ( S as − ( n − p ) T as )( e R ηαβγ − f ∆ e G ηαβγ )+ T as he g αβ ( e S γη + Ω e g γη ) − e g αγ ( e S βη + Ω e g βη ) i ,(IV) T ab Q ( e g, e S ) αβγδ = 0 and(V) e R · e R = ( f ∆ + Ω) Q ( e g, e R ) + Q ( e S, e R ) − ∆ Q ( e S, e G ) . Corollary 5.9.
If a warped product manifold M n = M p × f f M n − p satisfies R · R = Q ( S, R ) , then M = { x ∈ M : R | π ( x ) = 0 } ∪ { x ∈ M : ( e S − e κn − p e g ) | σ ( x ) = 0 } . Theorem 5.4.
If a warped product manifold M n = M p × f f M n − p satisfies R · R = Q ( S, R ) , theneither (i) the base M is flator (ii) the fiber f M is Einstein. Proof:
Since the condition R · R = Q ( S, R ) is a special case of R · R = L Q ( g, R ) + L Q ( S, R ), for L = 0 and L = 1. Therefore as L is nowhere zero, so from Theorem 5.3, we get our assertion. Corollary 5.10.
Let M n = M p × f f M n − p be a warped product manifold. Then M satisfies W · R = 0 if and only if the following conditions hold simultaneously: (I) R · R = κn ( n − Q ( g, R ) , (II) T ts (cid:0) R abct − T ac g bt + T bc g at (cid:1) = κn ( n − (cid:0) R abcs − T ac g bs + T bc g as (cid:1) , (III) T as (cid:16) e R ηαβγ − f ∆ e G ηαβγ (cid:17) + f T as e G ηαβγ = κn ( n − h g as e R ηαβγ + f ( T as − ∆ g as ) e G ηαβγ i ,(IV) e R · e R = f (cid:16) ∆ + κn ( n − (cid:17) Q ( e g, e R ) . Corollary 5.11.
Let M n = M p × f f M n − p be a warped product manifold. Then M satisfies W · R = L Q ( S, R ) if and only if the following conditions hold simultaneously: ( I ) R · R = κn ( n − Q ( g, R ) + L Q ( S, R ) − L ( n − p ) Q ( T, R ) , ( II ) f e g αβ (cid:0) T as T bc − T ac T bs + T ts R abct (cid:1) = κn ( n − f e g αβ (cid:0) R abcs − T ac g bs + T bc g as (cid:1) + L R abcs (cid:16) e S αβ + Ω e g αβ (cid:17) + L f e g αβ (cid:2) T bc (cid:0) S as − ( n − p ) T as (cid:1) − T ac (cid:0) S bs − ( n − p ) T bs (cid:1)(cid:3) , ( III ) T as (cid:16) e R ηαβγ − f ∆ e G ηαβγ (cid:17) + f T as e G ηαβγ = κn ( n − h g as e R ηαβγ + f ( T as − ∆ g as ) e G ηαβγ i + L ( S as − ( n − p ) T as )( e R ηαβγ − f ∆ e G ηαβγ )+ L T as he g αβ ( e S γη + Ω e g γη ) − e g αγ ( e S βη + Ω e g βη ) i , ( IV ) L T ab Q ( e g, e S ) αβγδ = 0 and ( V ) e R · e R = f (cid:16) ∆ + κn ( n − + Ω L (cid:17) Q ( e g, e R ) + L Q ( e S, e R ) − L ∆ Q ( e S, e G ) . Corollary 5.12.
Let M n = M p × f f M n − p be a warped product manifold. Then M satisfies P · R = 0 if and only if the following conditions hold simultaneously:(I) R · R = n − Q ( S − ( n − p ) T, R ) , (II) f e g αβ (cid:0) T as T bc − T ac T bs + T ts R abct (cid:1) = n − R abcs (cid:16) e S αβ + Ω e g αβ (cid:17) + n − f e g αβ (cid:2) T bc (cid:0) S as − ( n − p ) T as (cid:1) − T ac (cid:0) S bs − ( n − p ) T bs (cid:1)(cid:3) ,(III) T as (cid:16) e R ηαβγ − f ∆ e G ηαβγ (cid:17) + f T as e G ηαβγ = n − ( S as − ( n − p ) T as )( e R ηαβγ − f ∆ e G ηαβγ )+ n − T as he g αβ ( e S γη + Ω e g γη ) − e g αγ ( e S βη + Ω e g βη ) i ,(IV) T ab Q ( e g, e S ) αβγδ = 0 and(V) e R · e R = f (cid:0) ∆ + Ω n − (cid:1) Q ( e g, e R ) + n − Q ( e S, e R ) − n − ∆ Q ( e S, e G ) . Theorem 5.5.
If a warped product manifold M n = M p × f f M n − p satisfies P · R = 0 , then either(i) the base M is flat or (ii) the fiber f M is Einstein. Corollary 5.13.
Let M n = M p × f f M n − p be a warped product manifold. Then M satisfies thepseudosymmetric type condition (5.3) P · R = L Q ( g, R ) if and only if the following conditions hold simultaneously:(I) R · R = L Q ( g, R ) + n − Q ( S, R ) − n − pn − Q ( T, R ) , (II) f e g αβ (cid:0) T as T bc − T ac T bs + T ts R abct (cid:1) = L f e g αβ (cid:0) R abcs − T ac g bs + T bc g as (cid:1) + n − R abcs (cid:16) e S αβ + Ω e g αβ (cid:17) + fn − e g αβ (cid:2) T bc (cid:0) S as − ( n − p ) T as (cid:1) − T ac (cid:0) S bs − ( n − p ) T bs (cid:1)(cid:3) , N WARPED PRODUCT PSEUDOSYMMETRIC TYPE MANIFOLDS 13 (III) T as (cid:16) e R ηαβγ − f ∆ e G ηαβγ (cid:17) + f T as e G ηαβγ = L h g as e R ηαβγ + f ( T as − ∆ g as ) e G ηαβγ i + n − ( S as − ( n − p ) T as )( e R ηαβγ − f ∆ e G ηαβγ )+ n − T as he g αβ ( e S γη + Ω e g γη ) − e g αγ ( e S βη + Ω e g βη ) i ,(IV) T ab Q ( e g, e S ) αβγδ = 0 and(V) e R · e R = (cid:0) f ∆ + f L + Ω n − (cid:1) Q ( e g, e R ) + n − Q ( e S, e R ) − n − ∆ Q ( e S, e G ) . Theorem 5.6.
If a warped product manifold M n = M p × f f M n − p satisfies P · R = L Q ( g, R ) , theneither (i) the base M is flat or (ii) the fiber f M is Einstein. ExamplesExample 1:
Consider the warped product M = M × f f M , where M is an open interval of R with the metric ds = a (1+ x ) ( dx ) in local coordinate x , f M is a 4-dimensional manifoldequipped with a semi-Riemannian metric d e s = − ( dx ) + e x ( x ) ( dx ) + 2 e x dx dx + e x ( dx ) in local coordinates ( x , x , x , x ) and the warping function f = ( x + 1) . We can easily evaluatethe local components of necessary tensors of f M . The non-zero components of the Riemann-Christoffel curvature tensor e R and the Ricci tensor e S of f M upto symmetry are e R = e R = − e x , − e R = e R = e x , e R = − e x ( x ) , e R = e x ( e x x − e x x + 1)and e S = 3 , e S = − e x ( x ) − , e S = e S = − e x . Scalar curvature of f M is ( − e R · e R , Q ( e g, e R ) and Q ( e S, e R ) are e R · e R = − e R · e R = e x , e R · e R = − e R · e R = 2 e x , − Q ( e g, e R ) = Q ( e g, e R ) = e x , − Q ( e g, e R ) = Q ( e g, e R ) = 2 e x , − Q ( e S, e R ) = − Q ( e S, e R ) = 2 Q ( e S, e R ) = 23 Q ( e S, e R ) = − Q ( e S, e R ) = 2 e x , Q ( e S, e R ) = − Q ( e S, e R ) = 4 e x . Then we can easily check that f M satisfies e R · e R = − Q ( e g, e R ), i.e., f M is a pseudosymmetric manifoldof constant type.Now by a straightforward calculation we can evaluate the components of various necessarytensors corresponding to M . The non-zero local components of the Riemann-Christoffel curvaturetensor R and the Ricci tensor S of M upto symmetry are R = a ( x + 1) a ( x ) + 2 ax + a + 1 , R = − ae x ( x + 1) ( x ) a ( x ) + 2 ax + a + 1 , R = R = − ae x ( x + 1) a ( x ) + 2 ax + a + 1 , R = ae x ( x + 1) ( x ) ,R = R = ae x ( x + 1) , R = − R = ae x ( x + 1) , − R = e x ( x + 1) ( ae x ( x ) + ae x ( x ) ( x ) + 2 ae x x ( x ) + 1)and S = 4 aa ( x ) + 2 ax + a + 1 , S = − a ( x + 1) ,S = 4 ae x ( x ) + 4 ae x ( x ) ( x ) + 8 ae x x ( x ) + 1 , S = S = 4 ae x ( x + 1) . The scalar curvature of M is 20 a . Now the non-zero components (upto symmetry) of R · R , Q ( g, R )and Q ( S, R ) are R · R = − R · R = ae x ( x + 1) a ( x ) + 2 ax + a + 1 , − R · R = R · R = ae x ( x + 1) , − R · R = R · R = 2 ae x ( x + 1) ,Q ( g, R ) = − Q ( g, R ) = e x ( x + 1) a ( x ) + 2 ax + a + 1 , − Q ( g, R ) = Q ( g, R ) = e x ( x + 1) , − Q ( g, R ) = Q ( g, R ) = 2 e x ( x + 1) and − Q ( S, R ) = Q ( S, R ) = a ( x + 1) a ( x ) + 2 ax + a + 1 , − Q ( S, R ) = − Q ( S, R ) = Q ( S, R ) = 14 Q ( S, R ) = − Q ( S, R ) = ae x ( x + 1) a ( x ) + 2 ax + a + 1 , Q ( S, R ) = Q ( S, R ) = − Q ( S, R ) = − Q ( S, R ) = 13 Q ( S, R ) = ae x ( x + 1) , − Q ( S, R ) = Q ( S, R ) = 6 ae x ( x + 1) . Then from the values of R · R and Q ( g, R ), we see that M satisfies R · R = aQ ( g, R ). Therefore M is a warped product pseudosymmetric manifold of constant type. In particular if a = 0, then M is a warped product semisymmetric manifold. Now R · R = aQ ( g, R ) ⇒ ( R − aG ) · R = 0 and W = R − κ × G = R − aG hence W · R = 0. Therefore M is semisymmetric type due to concircularcurvature tensor. Remark 6.1.
We know that fibers of a warped product manifold are totally umbilical submanifoldof it. In the above example, f M is a totally umbilical hypersurface of M for a = 0 and for a = 0 , the N WARPED PRODUCT PSEUDOSYMMETRIC TYPE MANIFOLDS 15 above example is an example of pseudosymmetric totally umbilical hypersurface of a semisymmetricmanifold.
Example 2: [26] Let M = R × f R be the 4-dimensional connected semi-Riemannian warpedproduct manifold, where the base metric is given by ds = (1 + 2 e x )( dx ) , the fiber metric is theusual Euclidean metric and the warping function f = (1 + 2 e x ). Hence the metric of M is givenby(6.1) ds = (1 + 2 e x )( dx ) + (1 + 2 e x ) (cid:2) ( dx ) + ( dx ) + ( dx ) (cid:3) . Therefore the non-zero components (upto symmetry) of R , S , κ and P are given by R = R = R = − e x e x + 1 , R = R = R = − e x e x + 1 ; S = 3 e x (2 e x + 1) , S = S = S = e x e x + 1 ; κ = 6 e x (1 + e x )(1 + 2 e x ) ;12 P = 12 P = 12 P = P = − P = P = − P = P = − P = − e x − e x e x + 3 . Using above we can easily calculate the non-zero components (upto symmetry) of R · R , Q ( g, R ), Q ( S, R ) and P · R as follows: R · R = R · R = − R · R = R · R = − R · R = − R · R = e x (cid:16) e x − (cid:17) (2 e x + 1) ; Q ( g, R ) = Q ( g, R ) = − Q ( g, R ) = Q ( g, R ) = − Q ( g, R ) = − Q ( g, R ) = e x (cid:16) e x − (cid:17) ; Q ( S, R ) = Q ( S, R ) = − Q ( S, R ) = Q ( S, R ) = − Q ( S, R ) = − Q ( S, R ) = e x (cid:16) e x − (cid:17) (2 e x + 1) ; P · R = P · R = − P · R = P · R = − P · R = − P · R = 2 e x (cid:16) e x − (cid:17) e x + 1) . In view of above results we see that M satisfies the following pseudosymmetric type conditions:(i) R · R = e x ( e x +1 ) Q ( g, R ) = Q ( S, R ),(ii) P · R = e x ( e x +1 ) Q ( g, R ) and(iii) R · R = Q ( g, R ) + (cid:20) − e − x (cid:16) e x + 1 (cid:17) (cid:21) Q ( S, R ).7.
Conclusions
In this present paper we have found out the necessary and sufficient condition for which awarped product semi-Riemannian manifold M n = M p × f f M n − p satisfies the pseudosymmetric type condition R · R = L Q ( g, R ) + L Q ( S, R ). It is shown that on each point x ∈ M of suchmanifold either R = 0 or T = L g or e S = e κ ( n − p ) e g . Moreover if L is nowhere zero, then eitherthe base M is flat or the fiber f M is Einstein. As a special case of the main theorem we getthe necessary and sufficient condition for which a warped product semi-Riemannian manifold M n = M p × f f M n − p satisfies the pseudosymmetric type condition (i) R · R = 0, (ii) R · R = L Q ( g, R ),(iii) R · R = L Q ( S, R ), (iv) R · R = Q ( S, R ), (v) P · R = 0, (vi) P · R = LQ ( g, R ).It is proved that the base of a semisymmetric warped product manifold is semisymmetric andfiber is pseudosymmetric, whereas both the base and fiber of a pseudosymmetric warped productmanifold are pseudosymmetric. It is also proved that on a warped product pseudosymmetric orsemisymmetric manifold due to projective curvature tensor either the base is flat or the fiberis Einstein. By using the fact that the fiber of a semisymmetric warped product manifold ispseudosymmetric, we have established an example which ensures that a pseudosymmetric manifoldis a totally umbilical hypersurface of a semisymmetric manifold. References [1] Adam´ow, A. and Deszcz, R.,
On totally umbilical submanifolds of some class of Riemannian manifolds , Demon-stratio Math., (1983), 39–59.[2] Bishop, R. L. and O’Neill, B., Manifolds of negative curvature , Trans. Amer. Math. Soc., (1969), 1–49.[3] Cartan, E.,
Sur une classe remarquable d’espaces de Riemannian , Bull. Soc. Math. France, (1926), 214–264.[4] Cartan, E., Le¸cons sur la g´eom´etrie des espaces de Riemann , 2nd ed., Gauthier Villars, Paris, .[5] Defever, F. and Deszcz, R.,
On semi-Riemannian manifolds satisfying the condition R · R = Q ( S, R ), Geometryand Topology of Submanifolds,
III , World Sci., 1991, 108–130.[6] Defever, F. and Deszcz, R.,
On warped product manifolds satisfying a certain curvature condition , Atti Accad.Peloritana Pericolanti, Cl. Sci. Fis. Mat. Nat., (1991), 213–236.[7] Deszcz, R., On pseudosymmetric spaces , Bull. Belg. Math. Soc., Ser. A, (1992), 1–34.[8] Deszcz R., On pseudosymmetric warped product manifolds , In: Dillen F, editor. Geometry and Topology ofSubmanifolds, V. River Edge, NJ, USA: World Scientific, 1993, 132–146.[9] Deszcz R, G logowska M, Hotlo´s M and Sawicz K.
A Survey on Generalized Einstein Metric Conditions , Ad-vances in Lorentzian Geometry, Proceedings of the Lorentzian Geometry Conference in Berlin, AMS/IP Studiesin Advanced Mathematics, , S.-T. Yau (series ed.), M. Plaue, A.D. Rendall and M. Scherfner (eds.), 2011,27–46.[10] Deszcz, R., G logowska, M., Hotlo´s, M. and Zafindratafa, G., On some curvature conditions of pseudosymmetrytype , Period, Math. Hungarica, (2015), 153–170.[11] Deszcz, R., G logowska, M., Je lowicki, J. and Zafindratafa, G.,
Curvature properties of some class of warpedproduct manifolds , Int. J. Geom. Meth. Mod. Phys., (2016), 1550135.[12] Deszcz, R., Haesen, S. and Verstraelen, L.,
On natural symmetries. Chapter 6 in “Topics in DifferentialGeometry” , Editors A. Mihai, I. Mihai and R. Miron, Editura Acad. Romˆane (2008).[13] Deszcz, R., Hotlo´s, M., Je lowicki, J., Kundu, H. and Shaikh, A. A.,
Curvature properties of G¨odel metric , IntJ. Geom. Method Mod. Phy., (2014), 1450025, 20 pages, DOI: 10.1142/S021988781450025X.[14] G logowska, M., Semi-Riemannian manifolds whose Weyl tensor is a Kulkarni-Nomizu square , Publ. Inst. Math.(Beograd) (N.S.), (2002), 95–106.
N WARPED PRODUCT PSEUDOSYMMETRIC TYPE MANIFOLDS 17 [15] Haesen, S and Verstraelen, L.,
Properties of a scalar curvature invariant depending on two planes , ManuscriptaMath., (2007), 59–72.[16] Haesen, S. and Verstraelen, L.,
On the sectional curvature of Deszcz , Anal. Stiint. Univ. “Al. I. Cuza” Iasi(SN) Matematica, (2007), 181–190.[17] Haesen, S. and Verstraelen, L., Natural intrinsic geometrical symmetries , SIGMA, (2009), 1–15.[18] Jahanara, B., Haesen, S., Petrovi´c-Torga˘sev, M. and Verstraelen, L., On the Weyl curvature of Deszcz, Publ.Math. Debrecen, (2009), 417–431.[19] Kru˘ckovi˘c, G. I.,
On semi-reducible Riemannian spaces (in Russian), Dokl. Akad. Nauk SSSR, (1957),862–865.[20] Shaikh, A.A., Deszcz, R., Hotlo´s, M., Je lowicki, J. and Kundu, H.,
On pseudosymmetric manifolds , Publ.Math. Debrecen, (2015), 433–456.[21] Shaikh, A.A. and Kundu, H.,
On weakly symmetric and weakly Ricci symmetric warped product manifolds ,Publ. Math. Debrecen, (2012), 487–505.[22] Shaikh, A.A. and Kundu, H.,
On equivalency of various geometric structures , J. Geom., (2014), 139–165,DOI: 10.1007/s00022-013-0200-4, arXiv:1301.7214v3 [math.DG], 31 Jul 2013.[23] Shaikh, A.A. and Kundu, H.,
On generlized Roter type manifolds , arXiv:1411.0841v1 [math.DG] 4 Nov 2014.[24] Shaikh, A.A. and Kundu, H.,
On warped product generalized Roter type manifolds , Balkan J. Geom. Appl., (2016), 82–95.[25] Shaikh, A. A. and Kundu, H.,
On curvature properties of Som-Raychaudhuri spacetime , To appear in J. Geom.[26] Shaikh, A.A. and Kundu, H.,
On some curvature restricted geometric structures for projective curvature tensor ,arXiv:1609.04749v1 [math.DG], 15 Sep 2016.[27] Shaikh, A. A., Kundu, H. and Ali, Md. S.,
On warped product super generalized recurrent manifolds , To appearin An. S¸tiint¸. Univ. Al. I. Cuza Ia¸si Mat. (N.S.).[28] Shaikh, A. A., Roy, I. and Kundu, H.,
On the existence of a generalized class of recurrent manifolds ,arXiv:1504.0253v1 [math.DG] 10 Apr 2015.[29] Shaikh, A. A., Roy, I. and Kundu, H.,
On some generalized recurrent manifolds , To appear in Bull. IranianMath. Soc.[30] Szab´o, Z. I.,
Structure theorems on Riemannian spaces satisfying R ( X, Y ) .R = 0, I, The local version, J. Diff.Geom., (1982), 531–582.[31] Szab´o, Z. I., Classification and construction of complete hypersurfaces satisfying R ( X, Y ) · R = 0, Acta Sci.Math., (1984), 321–348.[32] Szab´o, Z. I., Structure theorems on Riemannian spaces satisfying R ( X, Y ) · R = 0 , II, The global version , Geom.Dedicata, (1985), 65–108. Department of Mathematics,University of Burdwan, Golapbag,Burdwan-713104,West Bengal, India
E-mail address : [email protected], [email protected] E-mail address ::