aa r X i v : . [ m a t h . DG ] J un GENERALIZED KENMOTSU MANIFOLDS
Aysel TURGUT VANLI and Ramazan SARI
Abstract
In 1972, K. Kenmotsu studied a class of almost contact Riemannian manifolds.Later, such a manifold was called a Kenmotsu manifold. This paper, we studiedKenmotsu manifolds with (2 n + s )-dimensional s − contact metric manifold and thismanifold, we have called generalized Kenmotsu manifolds. Necessary and sufficientcondition is given for an almost s − contact metric manifold to be a generalizedKenmotsu manifold.We show that a generalized Kenmotsu manifold is a locallywarped product space. In addition, we study some curvature properties of gener-alized Kenmotsu manifolds. Moreover, we show that the ϕ -sectional curvature ofany semi-symmetric and projective semi-symmetric (2 n + s )-dimensional generalizedKenmotsu manifold is − s . Introduction In [22], K.Yano introduced the notion of a f − structure on a differentiable manifold M ,i.e., a tensor fields f of type (1 ,
1) and rank n satisfying f + f = 0 as a generalization ofboth (almost) contact (for s = 1) and (almost) complex structures (for s = 0). T M splitsinto two complementary subbundles L = Imϕ and M = kerϕ . The existence of which isequivalent to a reduction of the structural group of the tangent bundle to U ( n ) × O ( s ) [4] . H. Nakagawa in [16] and [17] introduced the notion of globally framed f-manifolds( called f-manifolds ), later developed and studied by Goldberg and Yano [10], [11], [12] . Awide class of globally frame f -manifolds was introduced in [4], by Blair according to thefollowing definition. A metric f -structure is said to be a K -structure if the fundamental Mathematics Subject Classification . Primary 53C15 ; Secondary 53C25, 53D10.
Key words and phrases . Kenmotsu manifolds, metric f -manifolds, s-contact metric manifolds, gen-eralized Kenmotsu manifolds, semi-symmetric, ricci semi-symmetric, projective semi-symmetric. eneralized Kenmotsu Manifolds X, Y ) = g ( X, ϕY ), for any vector fields X and Y on M , isclosed and the normality condition holds, that is; [ ϕ, ϕ ] + 2 P si =1 dη i ⊗ ξ i = 0 where [ ϕ, ϕ ]denotes the Nijenhuis torsion of ϕ . Some authors studeid f -structure [5], [7], [23] . TheRiemannian connection ∇ of a metric f − manifold satisfies the following formula [6], g (( ∇ X ϕ ) Y, Z ) = 3 d Φ( X, ϕY, ϕZ ) − d Φ( X, Y, Z ) + g ( N ( Y, Z ) , ϕX )+ s X i =1 { N ( Y, Z ) η i ( X ) + 2 dη i ( ϕY, X ) η i ( Z ) − dη i ( ϕZ, X ) η i ( Y ) } , (1)where the tensor fields N and N are defined by N = [ ϕ, ϕ ]+2 P si =1 dη i ⊗ ξ i , N ( X, Y ) =( L ϕX η i )( Y ) − ( L ϕY η i )( X ) respectively, which is by a simple computation can be rewrittenas: N ( X, Y ) = 2 dη i ( ϕX, Y ) − dη i ( ϕY, X ) . Let M be a (2 n + 1) dimensional differentiable manifold. M is called an almost contactmetric manifold if ϕ is (1 ,
1) type tensor field, ξ is vector field, η is 1 − form and g is acompatible Riemannian metric such that(2) ϕ = − I + η ⊗ ξ, η ( ξ ) = 1(3) g ( ϕX, ϕY ) = g ( X, Y ) − η ( X ) η ( Y )for all X, Y ∈ Γ( T M ) . In addition, we have(4) η ( X ) = g ( X, ξ ) , ϕ ( ξ ) = 0 , η ( ϕ ) = 0for all X, Y ∈ Γ( T M ) [6].To study manifolds with negative curvature, Bishop and O’Neill introduced the notionof warped product as a generalization of Riemannian product [3] . In 1960’s and 1970’s,when almost contact manifolds were studied as an odd dimensional counterpart of almostcomplex manifolds, the warped product was used to make examples of almost contactmanifolds [20] . In addition, S. Tanno classified the connected (2 n + 1) dimensional almostcontact manifold M whose automorphism group has maximum dimension ( n + 1) in [20] . For such a manifold, the sectional curvature of plane sections containing ξ is a constant,say c . Then there are three classes. i ) c >
0, M is homogeneous Sasakian manifold of constant holomorphic sectionalcurvature. ii ) c = 0, M is the global Riemannian product of a line or a circle with a K¨ a hlermanifold of constant holomorphic sectional curvature. iii ) c <
0, M is warped product space R × f C n . Kenmotsu obtained some tensorialequations to characterize manifolds of the third case. eneralized Kenmotsu Manifolds [15],
Kenmotsu studied a class of almost contact Riemannian manifold which satisfythe following two condition,( ∇ X ϕ ) Y = − η ( Y ) ϕX − g ( X, ϕY ) ξ (5) ∇ X ξ = X − η ( X ) ξ He showed normal an almost contact Riemannian manifold with (5) but not quasiSasakian hence not Sasakian. He was to characterize warped product space L × f C E n byan almost contact Riemannian manifold with (5). Moreover, he show that every point ofan almost contact Riemannian manifold with (5) has a neighborhood which is a warped( − ǫ, ǫ ) × f V where f ( t ) = ce t and V is K¨ahler.In 1981 [13], Janssens and Vanhecke, an almost contact metric manifold satisfiying this(5) is called a Kenmotsu manifold. After this definition, some authors studied Kenmotsumanifold [8], [14], [18], [19].
The paper is organized as follows: after a preliminary basic notions of s − contact metricmanifolds theory, in Section 2, we introduced generalized almost Kenmotsu manifoldsand generalized Kenmotsu manifolds. Necessary and sufficient condition is given for a s − contact metric manifold to be a generalized Kenmotsu manifold. The warped product L s × f V n provides an example. In section 3, some curvature properties are given. Insection 4, we studied Ricci curvature tensor. In section 5, we studied semi-symmetricproperties of generalized Kenmotsu manifolds. We show that the ϕ -sectional curvatureof any semi-symmetric and projective semi-symmetric (2 n + s )-dimensional generalizedKenmotsu manifold is − s . In [11], a (2 n + s ) − dimensional differentiable manifold M is called metric f − manifold ifthere exist an (1 ,
1) type tensor field ϕ , s vector fields ξ , . . . , ξ s , s η , . . . , η s anda Riemannian metric g on M such that(6) ϕ = − I + s X i =1 η i ⊗ ξ i , η i ( ξ j ) = δ ij (7) g ( ϕX, ϕY ) = g ( X, Y ) − s X i =1 η i ( X ) η i ( Y ) , for any X, Y ∈ Γ( T M ) , i, j ∈ { , . . . , s } . In addition, we have(8) η i ( X ) = g ( X, ξ i ) , g ( X, ϕY ) = − g ( ϕX, Y ) , ϕξ i = 0 , η i ◦ ϕ = 0 . eneralized Kenmotsu Manifolds X, Y ) = g ( X, ϕY ), for any
X, Y ∈ Γ( T M ), calledthe fundamental 2 -form .In what follows, we denote by M the distribution spanned by the structure vector fields ξ , . . . , ξ s and by L its orthogonal complementary distribution. Then, T M = L ⊕ M . If X ∈ M we have ϕX = 0 and if X ∈ L we have η i ( X ) = 0, for any i ∈ { , . . . , s } ; that is, ϕ X = − X .In a metric f -manifold, special local orthonormal basis of vector fields can be consid-ered. Let U be a coordinate neighborhood and E a unit vector field on U orthogonalto the structure vector fields. Then, from (6) − (8), ϕE is also a unit vector field on U orthogonal to E and the structure vector fields. Next, if it is possible, let E be a unitvector field on U orthogonal to E , ϕE and the structure vector fields and so on. Thelocal orthonormal basis { E , . . . , E n , ϕE , . . . , ϕE n , ξ , . . . , ξ s } , so obtained is called an f -basis . Moreover, a metric f -manifold is normal if[ ϕ, ϕ ] + 2 s X i =1 dη i ⊗ ξ i = 0 , where [ ϕ, ϕ ] is denoting the Nijenhuis tensor field associated to ϕ .In [21], let M a (2 n + s ) − dimensional metric f − manifold. If there exists 2-form Φsuch that η ∧ ... ∧ η s ∧ Φ n = 0 on M then M is called an almost s-contact metric manifold. A normal almost s-contact metric manifold is called an s-contact metric manifold.
As is known in Kenmotsu manifold dimkerϕ = 1, since kerϕ = sp { ξ } . It was to be dimkerϕ > Ken-motsu s -structure ; that is, an almost s − contact metric manifold M is called a Kenmotsu s − manifold if ( ∇ X ϕ ) Y = g ( ϕX, Y ) s X i =1 ξ i − ϕX s X i =1 η i ( Y )for any X, Y ∈ Γ( T M ) [1] . We will give their definition as a theorem in this paper.Afterwards in 2006, M. Falcitelli and A.M. Pastore introduced
Kenmotsu f.pk-manifold .In [9] , a metric f.pk -manifold M of dimension 2 n + s , s ≥
1, with f.pk − structure whichis a metrik f − structure with parallelizable kernel. ( ϕ, ξ i , η i , g ) is said to be a Kenmotsu f.pk -manifold if it is normal, the1-forms η i are closed and d Φ = 2 η ∧ Φ . They assumethat d Φ = 2 η i ∧ Φ for all i = 1 , , ..., s in the definition of Kenmotsu f.pk − manifold. eneralized Kenmotsu Manifolds η i are linearly independent and η i ∧ Φ = η j ∧ Φimplies η i = η j , then the condition on d Φ can be satisfied by a unique η i and they canassume that d Φ = 2 η ∧ Φ . It is clear that authors were equated 1-forms η , ..., η s , whichdual of ξ , ..., ξ s . Thus, they studied unique 1-form η . In this paper, all η , ..., η s − forms are unequaled at the definition of generalizedKenmotsu manifolds. Definition . Let M be an almost s − contact metric manifold of dimension (2 n + s ), s ≥
1, with ( ϕ, ξ i , η i , g ) . M is said to be a generalized almost Kenmotsu manifold if forall 1 ≤ i ≤ s, − forms η i are closed and d Φ = 2 s P i =1 η i ∧ Φ . A normal generalized almostKenmotsu manifold M is called a generalized Kenmotsu manifold.Now, we construct an example of generalized Kenmotsu manifold. Example . We consider (2 n + s ) − dimensional manifold M = ( ( x , ..., x n , y , ..., y n , z , ..., z s ) ∈ R n + s : s X α =1 z α = 0 ) We choose the vector fields X i = e − s P α =1 z α ∂∂x i , Y i = e − s P α =1 z α ∂∂y i , ξ α = ∂∂z α , ≤ i ≤ n, ≤ α ≤ s which are linearly indepent at each point of M. Let g be the Riemannian metric definedby g = e s P α =1 z α " n X i =1 ( dx i ⊗ dx i + dy i ⊗ dy i ) + s X α =1 η α ⊗ η α . Hence, { X , ..., X n , Y , ..., Y n , ξ , ..., ξ s } is an orthonormal basis. Thus, η α be the 1 − formdefined by η α ( X ) = g ( X, ξ α ) , α = 1 , ..., s for any vector field X on T M.
We defined the(1 ,
1) tensor field ϕ as ϕ ( X i ) = Y i , ϕ ( Y i ) = − X i , ϕ ( ξ α ) = 0 , ≤ i ≤ n, ≤ α ≤ s. The linearity property of ϕ and g yields that η α ( ξ β ) = δ αβ , ϕ X = − X + s X α =1 η α ( X ) ξ α ,g ( ϕX, ϕY ) = g ( X, Y ) − s X α =1 η α ( X ) η α ( Y ) , eneralized Kenmotsu Manifolds X , Y on M. Therefore,(
M, ϕ, ξ α , η α , g ) defines a metric f − manifold.We have Φ( X i, , Y i ) = − ∂∂x i , ∂∂y i ) = g ( ∂∂x i , ϕ ∂∂y i ) = − e s P α =1 z α and hence, we have Φ = − e s P α =1 z α n X i =1 dx i ∧ dy i Therefore, we get η ∧ ... ∧ η s ∧ Φ n = 0 on M . Thus ( M, ϕ, ξ α , η α , g ) is almost s − contactmanifold. Consequently, the exterior derivative d Φ is given by d Φ = 2 s X α =1 dz α ∧ ( − e s P α =1 z α ) n X i =1 dx i ∧ dy i . Therefore, (
M, ϕ, ξ α , η α , g ) is a generalized almost Kenmotsu manifold. It can be seenthat ( M, ϕ, ξ α , η α , g ) is normal. So, it is a generalized Kenmotsu manifold. Moreover, weget [ X i , ξ α ] = X i , [ Y i , ξ α ] = Y i , [ X i , X j ] = 0 , [ X i , Y i ] = 0 , [ X i , Y j ] = 0[ Y i , Y j ] = 0 , ≤ i, j ≤ n, ≤ α ≤ s. The Riemannian connection ∇ of the metric g is given by2 g ( ∇ X Y, Z ) = Xg ( Y, Z ) +
Y g ( Z, X ) − Zg ( X, Y )+ g ([ X, Y ] , Z ) − g ([ Y, Z ] , X ) + g ([ Z, X ] , Y ) . Using the Koszul’s formula, we obtain ∇ X i X i = s X α =1 ξ α , ∇ Y i Y i = s X α =1 ξ α , ∇ X i X j = ∇ Y i Y j = ∇ X i Y i = ∇ X i Y j = 0 ∇ X i ξ α = X i , ∇ Y i ξ α = Y i , ≤ i, j ≤ n, ≤ α ≤ s. We construct an example of generalized Kenmotsu manifold for 7 − dimensional. Example . Let n = 2 and s = 3. The vector fields e = f ( z , z , z ) ∂∂x + f ( z , z , z ) ∂∂y ,e = − f ( z , z , z ) ∂∂x + f ( z , z , z ) ∂∂y , eneralized Kenmotsu Manifolds e = f ( z , z , z ) ∂∂x + f ( z , z , z ) ∂∂y ,e = − f ( z , z , z ) ∂∂x + f ( z , z , z ) ∂∂y ,e = ∂∂z , e = ∂∂z , e = ∂∂z where f and f are given by f ( z , z , z ) = c e − ( z + z + z ) cos( z + z + z ) − c e − ( z + z + z ) sin( z + z + z ) ,f ( z , z , z ) = c e − ( z + z + z ) cos( z + z + z ) + c e − ( z + z + z ) sin( z + z + z )for nonzero constant c , c . It is obvious that { e , e , e , e , e , e , e } are linearly indepen-dent at each point of M . Let g be the Riemannian metric given by g = 1 f + f
22 2 X i =1 ( dx i ⊗ dx i + dy i ⊗ dy i ) + dz ⊗ dz + dz ⊗ dz + dz ⊗ dz , where { x , y , x , y , z , z , z } are standard coordinates in R . Let η , η and η be the1 − form defined by η ( X ) = g ( X, e ), η ( X ) = g ( X, e ) and η ( X ) = g ( X, e ), respec-tively, for any vector field X on M and φ be the (1 ,
1) tensor field defined by ϕ ( e ) = e , ϕ ( e ) = − e , ϕ ( e ) = e , ϕ ( e ) = − e ,ϕ ( e = ξ ) = 0 , ϕ ( e = ξ ) = 0 , ϕ ( e = ξ ) = 0 . Therefore, the essential non-zero component of Φ isΦ( ∂∂x i , ∂∂y i ) = − f + f = − e z + z + z ) c + c , i = 1 , − e z + z + z ) c + c
22 2 X i =1 dx i ∧ dy i . Thus, we have η ∧ ... ∧ η s ∧ Φ n = 0 on M . Consequently, the exterior derivative d Φ isgiven by d Φ = − e z + z + z ) c + c ( dz + dz + dz ) ∧ X i =1 dx i ∧ dy i . Since η = dz , η = dz and η = dz , we find d Φ = 2( η + η + η ) ∧ Φ . In addition, Nijenhuis tersion of ϕ is equal to zero. eneralized Kenmotsu Manifolds Theorem . Let ( M, ϕ, ξ i , η i , g ) be an almost s -contact metric manifold. M is ageneralized Kenmotsu manifold if and only if (9) ( ∇ X ϕ ) Y = s X i =1 (cid:8) g ( ϕX, Y ) ξ i − η i ( Y ) ϕX (cid:9) for all X, Y ∈ Γ( T M ) , i ∈ { , , ..., s } , where ∇ is Riemannian connection on M. Proof.
Let M be a generalized Kenmotsu manifold. From (1), (6) , (7) and (8) forall X, Y ∈ Γ( T M ) , we have g (( ∇ X ϕ ) Y, Z ) = 3 ( s X i =1 ( η i ∧ Φ)(
X, ϕY, ϕZ ) − s X i =1 ( η i ∧ Φ)(
X, Y, Z ) ) = s X i =1 {
13 ( − η i ( X )Φ( ϕY, ϕZ ) + η i ( ϕY )Φ( ϕZ, X ) + η i ( ϕZ )Φ( X, ϕY )) −
13 ( − η i ( X )Φ( Y, Z ) + η i ( Y )Φ( Z, X ) + η i ( Z )Φ( X, Y )) } = s X i =1 (cid:8) − η i ( X ) g ( ϕY, ϕ Z ) + η i ( X ) g ( Y, ϕZ ) − η i ( Y ) g ( Z, ϕX ) − η i ( Z ) g ( X, ϕY ) (cid:9) = s X i =1 (cid:8) − η i ( Y ) g ( Z, ϕX ) − η i ( Z ) g ( X, ϕY ) (cid:9) = g ( s X i =1 (cid:8) g ( ϕX, Y ) ξ i − η i ( Y ) ϕX (cid:9) , Z ) . Conversely, firstly, using (9) and (8), we get ϕ ∇ X ξ j = s − X i =1 (cid:8) g ( ϕX, ξ j ) ξ i − η i ( ξ j ) ϕX (cid:9) hence, we get ϕ ∇ X ξ j = ϕ X. Therefore, we have ∇ X ξ j = − ϕ X. On the other hand, we get dη i ( X, Y ) = 12 { g ( Y, − ϕ X ) − g ( X, − ϕ Y ) } = 0for all X, Y ∈ Γ( T M ) , i ∈ { , , ..., s } . In addition, we know3 d Φ( X, Y, Z ) = Xg ( Y, ϕZ ) − Y g ( X, ϕZ ) − Zg ( X, ϕY ) − g ([ X, Y ] , ϕZ )+ g ([ X, Z ] , ϕY ) − g ([ Y, Z ] , ϕX )= g ( Y, ∇ X ϕZ − ϕ ∇ X Z ) − g ( X, ∇ Y ϕZ − ϕ ∇ Y Z ) + g ( X, ∇ Z ϕY − ϕ ∇ Z Y ) . eneralized Kenmotsu Manifolds d Φ( X, Y, Z ) = s X i =1 { g ( ϕX, Z ) g ( Y, ξ i ) − η i ( Z ) g ( Y, ϕX ) − g ( ϕY, Z ) g ( X, ξ i ) + η i ( Z ) g ( X, ϕY )+ g ( ϕZ, Y ) g ( X, ξ i ) − η i ( Y ) g ( X, ϕZ ) } = 2 s X i =1 { Φ( Z, X ) η i ( Y ) + Φ( X, Y ) η i ( Z ) + Φ( Y, Z ) η i ( X ) } . Then, we have, d Φ = 2 s X i =1 η i ∧ Φ . Moreover, the Nijenhuis torsion of ϕ is obtained N ϕ ( X, Y ) = ϕ − s X i =1 { g ( ϕX, Y ) ξ i − η i ( Y ) ϕX } + s X i =1 { g ( ϕY, X ) ξ i − η i ( X ) ϕY } ! + s X i =1 { g ( ϕ X, Y ) ξ i − η i ( Y ) ϕ X } − s X i =1 { g ( ϕ Y, X ) ξ i − η i ( X ) ϕ Y } = 0 . Hence, we have [ ϕ, ϕ ] + 2 s X i =1 dη i ⊗ ξ i = 0 . The proof is completed.
Corollary . Let M be a (2 n + s ) -dimensional generalized Kenmotsu manifoldwith structure ( ϕ, ξ i , η i , g ) . Then we have (10) ∇ X ξ j = − ϕ X for all X ∈ Γ( T M ) , i, j ∈ { , , ..., s } . Lemma . Let M be a (2 n + s ) -dimensional generalized Kenmotsu manifold withstructure ( ϕ, ξ i , η i , g ) . Then we have i ) ∇ ξ j ϕ = 0 , ∇ ξ j ξ i = 0 ii )( L ξ i ϕ ) X = 0 , ( L ξ i η j ) X = 0(11) iii )( L ξ i g )( X, Y ) = 2 { g ( X, Y ) − s X i =1 η i ( X ) η i ( Y ) } for all X ∈ Γ( T M ) , i, j ∈ { , , ..., s } . eneralized Kenmotsu Manifolds Theorem . Let M be a (2 n + s ) -dimensional generalized Kenmotsu manifold withstructure ( ϕ, ξ i , η i , g ) . Then we have (12) ( ∇ X η i ) Y = g ( X, Y ) − s X j =1 η j ( X ) η j ( Y )for all X, Y ∈ Γ( T M ) , i ∈ { , , ..., s } . Proof.
Using (8) and (10) we get the desired result.We have below the corollary in case s = 1. Corollary . Let ( M n +1 , ϕ, ξ, η, g ) be an almost contact metric manifold. M isa Kenmotsu manifold if and only if ( ∇ X ϕ ) Y = g ( ϕX, Y ) ξ − η ( Y ) ϕX for all X, Y ∈ Γ( T M ) , i ∈ { , , ..., s } , where ∇ is Riemannian connection on M [15]. Theorem . Let F be a K¨ahler manifold f ( t ) = ke s P i =1 t i be a function on R s , and kbe a non-zero constant. Then the warped product space M = R s × f F have a generalizedKenmotsu manifold. Proof.
Let (
F, J, G ) be a K¨ahler manifold and consider M = R s × f F, with co-ordinates ( t , ..., t s , x , ..., x n ). We define ϕ tensor field, 1-form η i , vector field ξ i andRiemannian metric tensor g on M as follows: ϕ ( ∂∂t i , U ) = (0 , J U ) ,η j ( ∂∂t i , U ) = δ ij , ξ i = ( ∂∂t i , g f = s X i =1 dt i ⊗ dt i + f π ∗ ( G )where f ( t ) = ke s P i =1 t i , U ∈ Γ( F ).Then ( M, ϕ, η i , ξ i , g f ) defines s − contact metric manifold. Now let us show that thismanifold is a generalized Kenmotsu manifold.It is clear that η i are closed. Thus, we haveΦ( X, Y ) = g f ( X, ϕY ) = f π ∗ ( G ( X, J Y ))or Φ = f π ∗ (Ψ) eneralized Kenmotsu Manifolds a hler manifold. Hence, we get d Φ = 2 c s X i =1 e s P i =1 t i dt i ∧ π ∗ (Ψ) = 2 s X i =1 dt i ∧ Φ . Finally torsion tensor N ϕ of M is vanish, since η i are closed and N J = 0.Then ( M = R s × f F, ϕ, η i , ξ i , g f ) is a generalized Kenmotsu manifold. Example . ( R × f V , g f = P i =1 dt i ⊗ dt i + f G ) is warped product with coordinates( t , t , x , x , x , x ), where f = k e P i =1 t i . Take a orthonormal frame field { E , E , E , E } of V and { e , e } of R such that E = J E , E = J E . Then we obtain a local orthonor-mal field { E , E , E , E , E , E } of R × f V by E = ke − P i =1 t i E , E = ke − P i =1 t i E E = − ke − P i =1 t i E , E = − ke − P i =1 t i E E = ξ , E = ξ . Then R × f V is a generalized Kenmotsu manifold. Theorem . Let ( M n + s , ϕ, η i , ξ i , g ) be a generalized Kenmotsu manifold, V and L are K¨ahler and a flat manifold with locally coordinates ( x , ..., x n ) and ( t , ..., t s ) re-spectively. Then M a locally warped product L s × f V n where f ( t ) = ke s P i =1 t i and k anonzero positive constant. Proof.
We know that
T M = L ⊕ M . L is clearly integrable, since dη i = 0 . Then V integral manifold of L is totally umbilical because ∇ X ξ i = X. On the other hand[ ξ i , ξ j ] = 0 and ∇ ξ i ξ j = 0, M is integrable and L integral manifold is totally geodesic.We select J = ϕ | D such that J = − I, G = g | D . Then (
V, J, G ) is almost Hermitianmanifold. Also torsion tensor N J = N ϕ = 0 and using (9) , we get ( ∇ X J ) Y = 0 . Then(
V, J, G ) is K¨ahler manifold.Then M = L × f V is locally a warped product and metric is g f = s X i =1 dt i ⊗ dt i + f G. Its follows that ( L ξ i g f )( X, Y ) = 2 ξ i ( f ) f G ( X, Y )and using (11), we get ξ i ( f ) = f, i = 1 , ..., s. eneralized Kenmotsu Manifolds ∂f ( t , ..., t s ) ∂t i = f ( t , ..., t s ) , i = 1 , ..., s. Therefore, we obtained f ( t , ..., t s ) = ce s P i =1 t i where c is nonzero constant. Example . Let’s go back to the example 2.3. Let ( R , ϕ, η i , ξ i , g ) be a generalizedKenmotsu manifold where i = 1 , ,
3. Take a orthonormal frame field (cid:26) ∂∂z = ξ , ∂∂z = ξ , ∂∂z = ξ (cid:27) of R and e P i =1 z i c + c ( f e − f e ) , e P i =1 z i c + c ( f e + f e ) , e P i =1 z i c + c ( f e − f e ) , e P i =1 z i c + c ( f e + f e ) of R . Then R = R × R is product manifold, the structure by tensor ϕ and metrictensor g. R is the standard K¨ahler structure ( J, G ). Here the Riemannian metric g iswarped product metric g + cf G where g is the Euclidean metric of R , f is the function defined on R by f ( z , z , z ) = e P i =1 z i and c = 1 c + c . Theorem . Let M be a (2 n + s ) -dimensional generalized Kenmotsu manifold withstructure ( ϕ, ξ i , η i , g ) . Then we have (13) R ( X, Y ) ξ i = s X j =1 { η j ( Y ) ϕ X − η j ( X ) ϕ Y } for all X, Y ∈ Γ( T M ) , i ∈ { , , ..., s } . Proof.
Firstly, using (10) and (6) we get ∇ X ∇ Y ξ i = ∇ X Y + ϕ X s X j =1 η j ( Y ) − s X j =1 { η j ( ∇ X Y ) ξ j + g ( Y, − ϕ X ) ξ j } and ∇ [ X,Y ] ξ i = − ϕ ∇ X Y + ϕ ∇ Y X. eneralized Kenmotsu Manifolds R ( X, Y ) ξ i = ∇ X Y + ϕ X s X j =1 η j ( Y ) − s X j =1 { η j ( ∇ X Y ) ξ j + g ( Y, − ϕ X ) ξ j }−∇ Y X − ϕ Y s X j =1 η j ( X ) + s X j =1 { η j ( ∇ Y X ) ξ j + g ( X, − ϕ Y ) ξ j } + ϕ ∇ X Y − ϕ ∇ Y X. From (6) desired result.
Corollary . Let M be a (2 n + s ) -dimensional generalized Kenmotsu manifoldwith structure ( ϕ, ξ i , η i , g ) . Then we have (14) R ( X, ξ i ) Y = s X j =1 { η j ( Y ) ϕ X − g ( X, ϕ Y ) ξ j } (15) R ( X, ξ j ) ξ i = ϕ X, R ( ξ k , ξ j ) ξ i = 0 for all X, Y ∈ Γ( T M ) , i, j, k ∈ { , , ..., s } . Corollary . [15] Let M be a (2 n +1) -dimensional Kenmotsu manifold with struc-ture ( ϕ, ξ, η, g ) . Then we have R ( X, Y ) ξ = η ( Y ) X − η ( X ) YR ( X, ξ ) Y = g ( X, Y ) ξ − η ( Y ) X, R ( ξ, ξ ) ξ = 0 for all X, Y ∈ Γ( T M ) . Theorem . Let M be a (2 n + s ) -dimensional generalized Kenmotsu manifold withstructure ( ϕ, ξ i , η i , g ) . Then we have ( ∇ Z R )( X, Y, ξ i ) = sg ( Z, X ) Y − sg ( Z, Y ) X − R ( X, Y ) Z + s s X h =1 η h ( Z ) { η h ( Y ) X − η h ( X ) Y } + s X l =1 η l ( Z ) R ( X, Y ) ξ l for all X, Y ∈ Γ( T M ) , i ∈ { , , ..., s } . Proof.
Using (10) and (13), we have( ∇ Z R )( X, Y, ξ i ) = ∇ Z { s X j =1 { η j ( X ) Y − η j ( Y ) X }} − s X j =1 { η j ( ∇ Z X ) Y − η j ( Y ) ∇ Z X }− s X j =1 { η j ( X ) ∇ Z Y − η j ( ∇ Z Y ) X } − R ( X, Y ) ϕ Z. eneralized Kenmotsu Manifolds ∇ Z R )( X, Y, ξ i ) = s X j =1 { g ( X, ∇ Z ξ j ) Y − g ( Y, ∇ Z ξ j ) X } − R ( X, Y ) Z + s X k =1 η k ( Y ) R ( X, Y ) ξ k . The proof competes from (6) and (10).
Corollary . Let M be a (2 n + s ) -dimensional generalized Kenmotsu manifoldThen we have ( ∇ Z R )( X, Y, ξ i ) = sg ( Z, X ) Y − sg ( Z, Y ) X − R ( X, Y ) Z, Z ∈ L ( ∇ ξ j R )( X, Y, ξ i ) = 0 for all X, Y ∈ Γ( T M ) , i ∈ { , , ..., s } . Corollary . [15] Let M be a (2 n +1) -dimensional Kenmotsu manifold with struc-ture ( ϕ, ξ, η, g ) . Then we have ( ∇ Z R )( X, Y, ξ ) = g ( Z, X ) Y − g ( Z, Y ) X − R ( X, Y ) Z, Z ∈ L and ( ∇ ξ R )( X, Y, ξ ) = 0 . Corollary . Let ( M, ϕ, ξ i , η i , g ) be a (2 n + s ) -dimensional locally-symmetric gen-eralized Kenmotsu manifold. Then we have R ( X, Y ) Z = s { g ( Z, X ) Y − g ( Z, Y ) X } . Corollary . [15] Let M be a (2 n + 1) -dimensional Kenmotsu manifold with struc-ture ( ϕ, ξ, η, g ) . If M is a locally symmetric then we have R ( X, Y ) Z = g ( Z, X ) Y − g ( Z, Y ) X. Corollary . The ϕ − sectional curvature of any locally symmetric generalized Ken-motsu manifold ( M, ϕ, ξ i , η i , g ) is equal to − s . In this case s = 1, we obtain that the ϕ − sectional curvature of any locally symmetricKenmotsu manifold ( M, ϕ, ξ, η, g ) is equal to − [15]. Theorem . Let M be a (2 n + s ) -dimensional generalized Kenmotsu manifold withstructure ( ϕ, ξ i , η i , g ) . Then we have R ( X, Y ) ϕZ − ϕR ( X, Y ) Z = g ( Y, Z ) ϕX − g ( X, Z ) ϕY − g ( Y, ϕZ ) X + g ( X, ϕZ ) YR ( ϕX, ϕY ) Z = R ( X, Y ) Z + g ( Y, Z ) X − g ( X, Z ) Y + g ( Y, ϕZ ) ϕX − g ( X, ϕZ ) ϕY for all X, Y ∈ Γ( T M ) , i ∈ { , , ..., s } . eneralized Kenmotsu Manifolds Theorem . Let M be a (2 n + s ) -dimensional generalized Kenmotsu manifold withstructure ( ϕ, ξ i , η i , g ) . Then we have (16) S ( X, ξ i ) = − n s X j =1 η j ( X ) for all X, Y ∈ Γ( T M ) , i ∈ { , , ..., s } . Proof. If { E , E , ..., E n + s } are local orthonormal vector fields, then S ( X, Y ) = n + s P k =1 g ( R ( E k , X ) Y, E k ) defines a global tensor field S of type (0 , S ( X, ξ i ) = n X k =1 g ( s X j =1 { η j ( X ) ϕ E k − η j ( E k ) ϕ X } , E k ) + s X k =1 g ( − ϕ X, ξ k )= s X j =1 η j ( X ) n X k =1 g ( ϕ E k , E k ) . In this case s = 1 we have S ( X, ξ ) = − nη ( X ) in [15]. Corollary . Let M be a (2 n + s ) -dimensional generalized Kenmotsu manifoldwith structure ( ϕ, ξ i , η i , g ) . Then we have (17) S ( ξ k , ξ i ) = − n for all X, Y ∈ T M, i, k ∈ { , , ..., s } . Theorem . Let M be a (2 n + s ) dimensional generalized Kenmotsu manifold withstructure ( ϕ, ξ i , η i , g ) . Then we have (18) S ( ϕX, ϕY ) = S ( X, Y ) + 2 n s X i =1 η i ( X ) η i ( Y ) for all X, Y ∈ Γ( T M ) , i ∈ { , , ..., s } . Proof.
We can put X = X + s X i =1 η i ( X ) ξ i and Y = Y + s X i =1 η i ( Y ) ξ i eneralized Kenmotsu Manifolds X , Y ∈ L . Then from (16) and (17) we have, S ( X, Y ) = S ( X , Y ) + s X i =1 η i ( Y ) η i ( X ) + s X i =1 η i ( X ) η i ( Y ) + s X i =1 η i ( X ) η i ( Y ) S ( ξ i , ξ i )= S ( X , Y ) − n s X i =1 η i ( X ) η i ( Y ) . Since ϕX, ϕY ∈ L we get S ( X , Y ) = S ( ϕX, ϕY ) which implies the desired result.Considering s = 1 in [14], we deduce S ( ϕX, ϕY ) = S ( X, Y ) + 2 nη ( X ) η ( Y ) . Theorem . Let M be a (2 n + s ) dimensional generalized Kenmotsu manifold withstructure ( ϕ, ξ i , η i , g ) . Then we have ( ∇ ϕX S )( ϕY, ϕZ ) = ( ∇ ϕX S )( Y, Z ) − s X i =1 η i ( Y ) { S ( X, ϕZ ) + 2 ng ( X, ϕZ ) }− s X i =1 η i ( Z ) { S ( X, ϕY ) + 2 ng ( X, ϕY ) } for all X, Y ∈ Γ( T M ) , i ∈ { , , ..., s } . Proof.
Using (9), we get( ∇ ϕX S )( ϕY, ϕZ ) = ∇ ϕX S ( Y, Z ) + 2 n s X i =1 { η i ( Y ) ∇ ϕX η i ( Z ) + η i ( Z ) ∇ ϕX η i ( Y ) } + s X i =1 {− S ( g ( ϕ X, Y ) ξ i − η i ( Y ) ϕ X, ϕZ ) − S ( ∇ ϕX Y, Z ) − nη i ( ∇ ϕX Y ) η i ( Z ) − S ( ϕY, g ( ϕ X, Z ) ξ i − η i ( Z ) ϕ X ) − S ( Y, ∇ ϕX Z ) − nη i ( Y ) η i ( ∇ ϕX Z ) } . From (6), (16) and (17) we have( ∇ ϕX S )( ϕY, ϕZ ) = ( ∇ ϕX S )( Y, Z ) + s X i =1 { nη i ( Y )( ∇ ϕX η i ) Z + 2 nη i ( Z )( ∇ ϕX η i ) Y − η i ( Y ) S ( X, ϕZ ) − η i ( Z ) S ( ϕY, X ) } . Corollary . Let M be a (2 n + s ) dimensional generalized Kenmotsu manifoldwith structure ( ϕ, ξ i , η i , g ) . Then we have ( ∇ X S )( ϕY, ϕZ ) = ( ∇ X S )( Y, Z ) + 2 n s X i =1 { g ( X, Y ) η i ( Z ) + g ( X, Z ) η i ( Y ) } + s X i =1 { η i ( Y ) S ( X, Z ) + η i ( Z ) S ( X, Y ) } for all X, Y ∈ Γ( T M ) , i ∈ { , , ..., s } . eneralized Kenmotsu Manifolds Definition . The Ricci tensor S of a (2 n + s )-dimensional generalized Kenmotsumanifold M is called η − parallel , if it satisfies( ∇ X S )( ϕY, ϕZ ) = 0for all vector fields X, Y and Z on M . Theorem . Let ( M, ϕ, ξ i , η i , g ) a (2 n + s ) -dimensional generalized Kenmotsu man-ifold. M has η − parallel if and only if ( ∇ X S )( Y, Z ) = − n s X i =1 { g ( X, Y ) η i ( Z ) + g ( X, Z ) η i ( Y ) }− s X i =1 { η i ( Y ) S ( X, Z ) + η i ( Z ) S ( X, Y ) } for all X, Y, Z ∈ Γ( T M ) , i ∈ { , , ..., s } . Corollary . [15] Let ( M, ϕ, ξ, η, g ) a (2 n + 1) -dimensional Kenmotsu manifold. M has η − parallel if and only if ( ∇ X S )( Y, Z ) = − n { g ( X, Y ) η ( Z ) + g ( X, Z ) η ( Y ) }− η ( Y ) S ( X, Z ) − η ( Z ) S ( X, Y ) for all X, Y, Z ∈ Γ( T M ) . With respect to the Riemannian connection ∇ of a generalized Kenmotsu manifold( M, ϕ, ξ i , η i , g ), we can prove: Theorem . The ϕ - sectional curvature of any semi-symmetric (2 n + s ) -dimensionalgeneralized Kenmotsu manifold ( M, ϕ, ξ i , η i , g ) is equal to − s . Proof.
Let X be a unit vector field. Since ( M, ϕ, ξ i , η i , g ) is semi-symmetric, then( R.R )( X, ξ i , X, ϕX, ϕX, ξ i ) = 0 , for any i, j ∈ { , , ..., s } . Expanding this formula from (7) and taking into account (14),we get R ( X, ϕX, ϕX, X ) = − s, which completes the proof. eneralized Kenmotsu Manifolds s = 1, by using the Theorem 11 we obtain that a semi-symmetric Kenmotsu manifolds is constant curvature equal to − Theorem . Let ( M, ϕ, ξ i , η i , g ) be a (2 n + s ) dimensional Ricci semi-symmetricgeneralized Kenmotsu manifold. Then its Ricci tensor field S respect the Riemannianconnection satisfies (19) S ( X, Y ) = − n { sg ( ϕX, ϕY ) + s X i,j =1 η i ( X ) η j ( Y ) } for any X, Y ∈ Γ( T M ) . Proof.
Since (
M, ϕ, ξ i , η i , g ) is Ricci semi-symmetric, then S ( R ( X, ξ i ) ξ j , Y ) + S ( ξ j , R ( X, ξ i ) Y ) = 0 , for any X, Y ∈ Γ( T M ) and i, j ∈ { , , ..., s } . Now, from (14) , (15) and (16) we get thedesired result.In this case s = 1 we have following the corollary. Corollary . Any Ricci semi-symmetric (2 n +1) − dimensional Kenmotsu manifoldis an Einstein manifold. Proof.
Considering s = 1 in (19), we deduce S ( X, Y ) = − ng ( X, Y )for any
X, Y ∈ Γ( T M ) . For the Weyl projective curvature tensor field P , the weyl projective curvature tensor P of a (2 n + s )-dimensional generalized Kenmotsu manifold M is given by P ( X, Y ) Z = R ( X, Y ) Z − n + s − { S ( Y, Z ) X − S ( X, Z ) Y } where R is curvature tensor and S is the ricci curvature tensor of M , we have the followingtheorem. Theorem . The ϕ - sectional curvature of any projectively semi-symmetric gener-alized Kenmotsu manifold ( M, ϕ, ξ i , η i , g ) is equal to − s . Proof.
Let X be a unit vector field. Then, from (6) and taking into account (14)and (16) we have( R.P )( X, ξ i , X, ϕX, ϕX, ξ j ) = ( R.R )( X, ξ i , X, ϕX, ϕX, ξ j ) . This completes the proof from the
Theorem 5.1 . Corollary . Let ( M, ϕ, ξ, η, g ) a (2 n + 1) -dimensional Kenmotsu manifold. The ϕ - sectional curvature of any projectively semi-symmetric Kenmotsu manifold if and onlyif M is an Einstein manifold. eneralized Kenmotsu Manifolds References [1]
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Department of MathematicsFaculty of SciencesGazi UniversityANKARA 06500, TURKEI