New curvature tensors along Riemannian submersions
aa r X i v : . [ m a t h . G M ] J u l New curvature tensors along Riemannian sub-mersions
Mehmet Akif Akyol and G¨ulhan Ayar
Abstract.
In 1966, B. O’Neill [The fundamental equations of a sub-mersion, Michigan Math. J., Volume 13, Issue 4 (1966), 459-469.] ob-tained some fundamental equations and curvature relations betweenthe total space, the base space and the fibres of a submersion. In thepresent paper, we define new curvature tensors along Riemannian sub-mersions such as Weyl projective curvature tensor, concircular curvaturetensor, conharmonic curvature tensor, conformal curvature tensor and M − projective curvature tensor, respectively. Finally, we obtain some re-sults in case of the total space of Riemannian submersions has umbilicalfibres for any curvature tensors mentioned by the above. Mathematics Subject Classification (2010).
Primary 53C15, 53B20.
Keywords.
Riemannian submersion, Weyl projective curvature tensor, M -projective curvature tensor, concircular curvature tensor, conformalcurvature tensor, conharmonic curvature tensor.
1. Introduction and Preliminaries
In differential geometry, an important tool to define the curvature of n − dimensional spaces (such as Riemannian manifolds) is the Riemannian cur-vature tensor. The tensor has played an important role both general rela-tivty and gravity. In this manner, Mishra in [10] defined some new curvaturetensors on Riemannian manifolds such as concircular curvature tensor, con-harmonic curvature tensor, conformal curvature tensor, respectively. Takinginto account the paper of Mishra, Pokhariyal and Mishra defined the Welyprojective curvature tensor on Riemannian manifolds [13]. Afterwards, Ojhadefined M − projective curvature tensor [11].Curvature tensors also play an important role in physics. Relevance inphysics of the tensors considered in our work and some of the importantstudies that focus on the geometric properties of curvature tensors are; in1988 conditions of conharmonic curvature tensor of Kaehler hypersurfaces in M. A. Akyol and G. Ayarcomplex space forms has been analysed by M. Doric et al. [19]. By the way,the relativistic significance of concircular curvature tensor has been studiedby Ahsan [18] and this tensor has been explored in that study. Finaly in 2018based on Einsteins geodesic postulate, projectively related connections on aspace-time manifold and the closely related Weyl projective tensor has beenexamined detailed by G. Hall [17].Riemannian submersion appears to have been studied and its differentialgeometry has first defined by O’Neill 1966 and Gray 1967 [12]. We note thatRiemannian submersions have been studied widely not only in mathematics,but also in theoretical pyhsics because of their applications in the Yang-Millstheory, Kaluza Klein theory, super gravity, relativity and superstring theories(see [2], [3], [7], [8], [9], [16]). Most of the studies related to Riemannian sub-mersion can be found in the books ([5], [15]). In 1966, B. O’Neill has defined apaper related to some fundamental equations of a submersion. In that paper,he has given some curvature relations on Riemannian submersions.In this study, in addition to the curvature relations previously defined onRiemannian submersion, we investigate new curvature tensors on a Riemann-ian submersion and the curvature properties of these tensors. In the presentpaper, in the first part of our study, the basic definitions and theorems thatwe will use throughout the paper are given. In sections 2-6 include the Weylprojective curvature tensor, concircular curvature tensor, conharmonic cur-vature tensor, conformal curvature tensor and M -projective curvature tensorrelations for a Riemannian submersion respectively. Also various results areobtained by examining the conditions for having total umbilical fibers.Now, we will give the basic definitions and theorems without proofs thatwe will use throughout the paper. Definition 1.1.
Let (
M, g ) and (
N, g N ) be Riemannian manifolds, where- dim ( M ) > dim ( N ). A surjective mapping π : ( M, g ) → ( N, g N ) is calleda Riemannian submersion [12] if: (S1)
The rank of π equals dim ( N ).In this case, for each q ∈ N , π − ( q ) = π − q is a k -dimensional submanifoldof M and called a fiber , where k = dim ( M ) − dim ( N ) . A vector field on M is called vertical (resp. horizontal ) if it is always tangent (resp. orthogo-nal) to fibers. A vector field X on M is called basic if X is horizontal and π -related to a vector field X ∗ on N, i.e. , π ∗ ( X p ) = X ∗ π ( p ) for all p ∈ M, where π ∗ is derivative or differential map of π. We will denote by V and H the projections on the vertical distribution kerπ ∗ , and the horizontal dis-tribution kerπ ⊥∗ , respectively. As usual, the manifold ( M, g ) is called totalmanifold and the manifold (
N, g N ) is called base manifold of the submersion π : ( M, g ) → ( N, g N ). (S2) π ∗ preserves the lengths of the horizontal vectors.This condition is equivalent to say that the derivative map π ∗ of π , restrictedto kerπ ⊥∗ , is a linear isometry.If X and Y are the basic vector fields, π -related to X N , Y N , we have thefollowing facts:ew curvature tensors along Riemannian submersion 31. g ( X, Y ) = g N ( X N , Y N ) ◦ π ,2. h [ X, Y ] is the basic vector field π -related to [ X N , Y N ],3. h ( ∇ X Y ) is the basic vector field π -related to ∇ N X N Y N ,for any vertical vector field V , [ X, V ] is the vertical.The geometry of Riemannian submersions is characterized by O’Neill’stensors T and A , defined as follows: T E F = v ∇ vE hF + h ∇ vE vF, (1.1) A E F = v ∇ hE hF + h ∇ hE vF (1.2)for any vector fields E and F on M, where ∇ is the Levi-Civita connec-tion, v and h are orthogonal projections on vertical and horizontal spaces,respectively.We now recall the following curvature relations for a Riemannian sub-mersion from [5] and [12]. Theorem 1.2. ( M, g ) and ( G, g ′ ) Riemannian manifolds, π : ( M, g ) → ( G, g ′ ) a Riemannian submersion and R M , R G and ˆ R be Riemannian curvaturetensors of M, G and ( π − ( x ) , ˆ g x ) fibre respectively. In this case, there are thefollowing equations for any U, V, W, F ∈ χ v ( M ) and X, Y, Z, H ∈ χ h ( M ) g ( R M ( X, Y ) Z, H ) = g ( R G ( X, Y ) Z, H ) + 2 g ( A X Y, A Z H ) − g ( A Y Z, A X H ) + g ( A X Z, A Y H ) , (1.3) g ( R M ( X, Y ) Z, V ) = − g (( ∇ Z A ) X Y, V ) − g ( A X Y, T V Z )+ g ( A Y Z, T V X ) − g ( A X Z, T V Y ) , (1.4) g ( R M ( X, Y ) V, W ) = g (( ∇ V A ) X Y, W ) − g (( ∇ W A ) X Y, V )+ g ( A X V A Y W ) + g ( A X W, A Y V ) − g ( T V X, T W Y ) + g ( T W X, T V Y ) , (1.5) g ( R M ( X, V ) Y, W ) = g (( ∇ X T ) V W, Y ) − g (( ∇ V A ) X Y, W ) − g ( T V X, T W Y ) + g ( A X V, A Y W ) , (1.6) g ( R M ( U, V ) W, X ) = g (( ∇ U T ) V W, X ) − g (( ∇ V T ) U W, X ) (1.7) and g ( R M ( U, V ) W, F ) = g ( ˆ R ( U, V ) W, F ) + g ( T U W, T V F ) − g ( T V W, T U F ) . (1.8) M. A. Akyol and G. Ayar Definition 1.3. [5] Let (
M, g ) be a Riemannian manifold and a local orthonor-mal frame of the vertical distribution ν is { U j } ≤ j ≤ r . Then N , the horizontalvector field on ( M, g ) is locally defined by N = r X j =1 T U j U j . Proposition 1.4.
Let ( M, g ) and ( G, g ′ ) Riemannian manifolds, π : ( M, g ) → ( G, g ′ ) a Riemannian submersion and { X i , U j } be a π -compatible frame.In this case, for any U, V ∈ χ v ( M ) and X, Y ∈ χ h ( M ) , the Ricci tensor S M satisfies the following equations [5] : ( i ) S M ( U, V ) = ˆ S ( U, V ) − g ( N, T U V ) (1.9)+ X i { g (( ∇ X i T ) U V, X i ) + g ( A X i U, A X i V ) } , ( ii ) S M ( X, Y ) = S G ( X ′ , Y ′ ) ◦ π + 12 { g ( ∇ X N, Y ) + g ( ∇ Y N, X ) } (1.10) − X i g ( A X X i , A Y X i ) − X j g ( T U j X, T U j Y ) , ( iii ) S M ( U, X ) = g ( ∇ U N, X ) − X j g ( ∇ U j T ) U j U, X ) (1.11)+ X i { g (( ∇ X i A ) X i X, U ) − g ( A X X i , T U X i ) } . Proposition 1.5. [5]
Let’s take the scalar curvatures of ( M, g ) , ( G, g ′ ) Rie-mannian manifolds and x ∈ G, π − ( x ) fibre with r M , r G and ˆ r , respectively.In a π : ( M, g ) → ( G, g ′ ) Riemannian submersion, ( M, g ) depends on the scalar curvature of the Rie-mannian manifold r G and the scalar curve of any lift ˆ r . In this case r M = ˆ r + r G ◦ π − || N || − || A || − || T || + 2 X i g ( ∇ X i N, X i ) . (1.12)
2. Weyl projective curvature tensor along a Riemanniansubmersion
In this section, we examine the Weyl projective curvature tensor relations be-tween the total space, the base space and fibres on a Riemannian submersion.We also give a corollary in case of the Riemannian submersion has totallyumbilical fibres.ew curvature tensors along Riemannian submersion 5
Definition 2.1. [10] Let take an n -dimensional differentiable manifold M withdifferentiability class C ∞ . In the n − dimensional space V n , the tensor P ∗ ( X, Y ) Z = R M ( X, Y ) Z − n − { S M ( Y, Z ) X − S M ( X, Z ) Y } . is called Weyl projective curvature tensor, where Ricci tensor of total spacedenoted by S M . Now, we have the following main theorem.
Theorem 2.2.
Let ( M, g ) and ( G, g ′ ) Riemannian manifolds, π : ( M, g ) → ( G, g ′ ) a Riemannian submersion and R M , R G and ˆ R be Riemannian curvaturetensors, S M , S G and ˆ S be Ricci tensors of M, G and the fibre respectively.Then for any
U, V, W, F ∈ χ v ( M ) and X, Y, Z, H ∈ χ h ( M ) , we have thefollowing relations for Weyl projective curvature tensor: g ( P ∗ ( X, Y ) Z, H )= g ( R G ( X, Y ) Z, H )+2 g ( A X Y, A Z H ) − g ( A Y Z, A X H )+ g ( A X Z, A Y H ) − n − ( g ( X, H ) (cid:20) S G ( Y ′ , Z ′ ) ◦ π + 12 ( g ( ∇ Y N, Z ) + g ( ∇ Z N, Y )) − X i g ( A Y X i , A Z X i ) − X j g ( T U j Y, T U j Z ) (cid:21) − g ( Y, H ) (cid:20) S G ( X ′ , Z ′ ) ◦ π + 12 ( g ( ∇ X N, Z ) + g ( ∇ Z N, X )) − X i g ( A X X i , A Z X i ) − X j g ( T U j X, T U j Z ) (cid:21)) ,g ( P ∗ ( X, Y ) Z, V ) = − g (( ∇ Z A ) X Y, V ) − g ( A X Y, T V Z ) + g ( A Y Z, T V X ) − g ( A X Z, T V Y ) ,g ( P ∗ ( X, Y ) V, W ) = g (( ∇ V A ) X Y, W ) − g (( ∇ W A ) X Y, V ) + g ( A X V, A Y W ) − g ( A X W, A Y V ) − g ( T V X, T W Y ) + g ( T W X, T V Y ) ,g ( P ∗ ( X, V ) Y, W ) = g (( ∇ X T ) V W, Y ) + g (( ∇ V A ) X Y, W ) − g ( T V X, T W Y )+ g ( A X V, A Y W ) ,g ( P ∗ ( U, V ) W, X ) = g (( ∇ U T ) V W, X ) − g (( ∇ V T ) U W, X ) M. A. Akyol and G. Ayar and g ( P ∗ ( U, V ) W, F ) = g ( ˆ R ( U, V ) W, F ) + g ( T U W, T V F ) − g ( T V W, T U F ) − n − ( g ( F, U ) (cid:20) ˆ S ( V, W ) − g ( N, T V W )+ X i ( g (( ∇ X i T ) V W, X i ) + g ( A X i V, A X i W )) (cid:21) − g ( F, V ) (cid:20) ˆ S ( U, W ) − g ( N, T U W )+ X i ( g (( ∇ X i T ) U W, X i ) + g ( A X i U, A X i W )) (cid:21)) . Proof.
We only give the proof of the 1 st equation of this theorem. The fol-lowing equations are obtained inner production with H to P ∗ and using (1.3)and (1.10) equations. g ( R M ( X, Y ) Z, H ) = g ( R G ( X, Y ) Z, H ) + 2 g ( A X Y, A Z H ) − g ( A Y Z, A X H )+ g ( A X Z, A Y H ) ,S M ( Y, Z ) = S G ( Y ′ , Z ′ ) ◦ π + 12 { g ( ∇ Y N, Z ) + g ( ∇ Z N, Y ) }− X i g ( A Y Y i , A Z Y i ) − X j ( T U j Y, T U j Z )and S M ( X, Z ) = S G ( Y ′ , Z ′ ) ◦ π + 12 { g ( ∇ X N, Z ) + g ( ∇ Z N, X ) }− X i g ( A X X i , A Z X i ) − X j ( T U j X, T U j Z ) . When these equations are substituted in P ∗ , the given result is obtained.Other equations are similarly proved by using Theorem 1.2 and Proposition1.4. (cid:3) Corollary 2.3.
Let π : ( M, g ) → ( G, g ′ ) be a Riemannian submersion, where ( M, g ) and ( G, g ′ ) Riemannian manifolds. If the Riemannian submersion hastotal umbilical fibres, that is N = 0 , then the Weyl projective curvature tensor ew curvature tensors along Riemannian submersion 7 is given by g ( P ∗ ( X, Y ) Z, H )= g ( R G ( X, Y ) Z, H ) + 2 g ( A X Y, A Z H ) − g ( A Y Z, A X H ) + g ( A X Z, A Y H ) − n − ( g ( X, H ) (cid:20) S G ( Y ′ , Z ′ ) ◦ π − X i g ( A Y X i , A Z X i ) − X j g ( T U j Y, T U j Z ) (cid:21) − g ( Y, H ) (cid:20) S G ( X ′ , Z ′ ) ◦ π − X i g ( A X X i , A Z X i ) − X j g ( T U j X, T U j Z ) (cid:21)) , and g ( P ∗ ( U, V ) W, F )= g ( ˆ R ( U, V ) W, F ) + g ( T U W, T V F ) − g ( T V W, T U F ) − n − ( g ( F, U ) (cid:20) ˆ S ( V, W ) + X i ( g (( ∇ X i T ) V W, X i ) + g ( A X i V, A X i W )) (cid:21) − g ( F, V ) (cid:20) ˆ S ( U, W ) + X i ( g (( ∇ X i T ) U W, X i ) + g ( A X i U, A X i W )) (cid:21)) . for any U, V, W, F ∈ χ v ( M ) and X, Y, Z, H ∈ χ h ( M ) ,
3. Concircular curvature tensor along a Riemanniansubmersion
In this section, curvature relations of concircular curvature tensor in a Rie-mannian submersion are examined. In particular, we show that the Riemann-ian submersion with concircular curvature tensor has no the totally umbilicalfibres.
Definition 3.1.
In the n − dimensional space V n , the tensor C ∗ ( X, Y, Z, H ) = R M ( X, Y, Z, H ) − r M n ( n −
1) [ g ( X, H ) g ( Y, Z ) − g ( Y, H ) g ( X, Z )] , is called concircular curvature tensor, where scalar tensor denoted by r M [10].Now, we have the following main theorem. Theorem 3.2.
Let ( M, g ) and ( G, g ′ ) Riemannian manifolds, π : ( M, g ) → ( G, g ′ ) a Riemannian submersion and R M , R G and ˆ R be Riemannian curvaturetensors, r M , r G and ˆ r be scalar curvature tensors of M, G and the fibre
M. A. Akyol and G. Ayar respectively. Then for any
U, V, W, F ∈ χ v ( M ) and X, Y, Z, H ∈ χ h ( M ) , wehave the following relations g ( C ∗ ( X, Y ) Z, H ) = g ( R G ( X, Y ) Z, H ) + 2 g ( A X Y, A Z H ) − g ( A Y Z, A X H ) + g ( A X Z, A Y H ) − r M n ( n − ( g ( Y, Z ) g ( X, H ) − g ( X, Z ) g ( Y, H ) ) ,g ( C ∗ ( X, Y ) Z, V ) = − g (( ∇ Z A ) X Y, V ) − g ( A X T, T V Z )+ g ( A Y Z, T V X ) − g ( A X Z, T V Y ) ,g ( C ∗ ( X, Y ) V, W ) = g (( ∇ V A ) X Y, W ) − g (( ∇ W A ) X Y, V ) + g ( A X V, A Y W ) − g ( A X W, A Y V ) − g ( T V X, T W Y ) + g ( T W X, T V Y ) ,g ( C ∗ ( X, V ) Y, W ) = g (( ∇ X T ) V W, Y ) + g (( ∇ V A ) X Y, W ) − g ( T V X, T W Y )+ g ( A X Y, A Y W ) − r M n ( n − {− g ( X, Y ) g ( V, W ) } ,g ( C ∗ ( U, V ) W, X ) = g (( ∇ U T ) V W, X ) − g (( ∇ V T ) U W, X ) and g ( C ∗ ( U, V ) W, F ) = g ( ˆ R ( U, V ) W, F ) + g ( T U W, T V F ) − g ( T V W, T U F ) − r M n ( n − ( g ( V, W ) g ( U, F ) − g ( U, W ) g ( V, F ) ) where r M = ˆ r + r G ◦ π − || A || − || T || . Proof.
Let’s prove the 2 nd equation of this theorem. Taking inner product C ∗ with V then we have g ( C ∗ ( X, Y ) Z, V ) = g ( R ( X, Y ) Z, V ) − r M n ( n − { g ( Y, Z ) g ( X, V ) − g ( X, Z ) } . Then using equation (1.4), we get g ( C ∗ ( X, Y ) Z, V ) = − g (( ∇ Z A ) X Y, V ) − g ( A X T, T V Z )+ g ( A Y Z, T V X ) − g ( A X Z, T V Y ) . which completes the proof of the second equation. Other equations are simi-larly proved by using Theorem 1.2, Proposition 1.4 and Proposition 1.5. (cid:3) Corollary 3.3.
Let π : ( M, g ) → ( G, g ′ ) be a Riemannian submersion, where ( M, g ) and ( G, g ′ ) Riemannian manifolds. Then the concircular curvaturetensor of Riemannian submersion has no total umbilical fibres. ew curvature tensors along Riemannian submersion 9
4. Conharmonic curvature tensor along a Riemanniansubmersion
In this section, curvature relations of conharmonic curvature tensor in a Rie-mannian submersion are examined.
Definition 4.1.
In the n − dimensional space V n , the tensor L ∗ ( X, Y, Z, H ) = R M ( X, Y, Z, H ) − n − g ( Y, Z ) Ric ( X, H ) − g ( X, Z ) Ric ( Y, H ) , is called conharmonic curvature tensor, where Ricci tensor denoted by Ric [10]. In a similar way, we have the following main theorem.
Theorem 4.2.
Let ( M, g ) and ( G, g ′ ) Riemannian manifolds, π : ( M, g ) → ( G, g ′ ) a Riemannian submersion and R M , R G and ˆ R be Riemannian curvaturetensors, S M , S G and ˆ S be Ricci tensors of M, G and the fibre respectively.Then for any
U, V, W, F ∈ χ v ( M ) and X, Y, Z, H ∈ χ h ( M ) , we have thefollowing relations g ( L ∗ ( X, Y ) Z, H )= g ( R G ( X, Y ) Z, H ) + 2 g ( A X Y, A Z H ) − g ( A Y Z, A X H ) + g ( A X Z, A Y H ) − n − ( g ( Y, Z ) (cid:20) S G ( X ′ , H ′ ) ◦ π + 12 ( g ( ∇ X N, H ) + g ( ∇ H N, X )) − X i g ( A X X i , A H X i ) − X j g ( T U j X, T U j H ) (cid:21) − g ( X, Z ) (cid:20) S G ( Y ′ , H ′ ) ◦ π + 12 ( g ( ∇ Y N, H ) + g ( ∇ H N, Y )) − X i g ( A Y X i , A H X i ) − X j g ( T U j Y, T U j H ) (cid:21) + g ( X, H ) (cid:20) S G ( Y ′ , Z ′ ) ◦ π + 12 ( g ( ∇ Y N, Z ) + g ( ∇ Z N, Y )) − X i g ( A Y X i , A Z X i ) − X j g ( T U j Y, T U j Z ) − g ( Y, H ) (cid:20) S G ( X ′ , Z ′ ) ◦ π + 12 ( g ( ∇ X N, Z ) + g ( ∇ Z N, X )) − X i g ( A X X i , A Z X i ) − X j g ( T U j X, T U j Z ) (cid:21)) , g ( L ∗ ( X, Y ) Z, V )= − g (( ∇ Z A ) X Y, V ) − g ( A X T, T V Z ) + g ( A Y Z, T V X ) − g ( A X Z, T V Y ) − n − ( g ( Y, Z ) (cid:20) g ( ∇ X N, V ) − X j g (( ∇ U j T ) U j X, V )+ X i ( g (( ∇ X i A ) X i X, V ) − g ( A V X i , T X X i )) (cid:21) − g ( X, Z ) (cid:20) g ( ∇ Y N, V ) − X j g (( ∇ U j T ) U j Y, V )+ X i ( g (( ∇ X i A ) X i Y, V ) − g ( A V X i , T Y X i )) (cid:21)) ,g ( L ∗ ( X, Y ) V, W ) = g (( ∇ V A ) X Y, W ) − g (( ∇ W A ) X Y, V ) + g ( A X V, A Y W ) − g ( A X W, A Y V ) − g ( T V X, T W Y ) + g ( T W X, T V Y ) ,g ( L ∗ ( X, V ) Y, W )= g (( ∇ X T ) V W, Y ) + g (( ∇ V A ) X Y, W ) − g ( T V X, T W Y ) + g ( A X Y, A Y W ) − n − (cid:26) − g ( V, W ) h S G ( X ′ , Y ′ ) ◦ π + 12 ( g ( ∇ X N, Y ) + g ( ∇ Y N, X )) − X i g ( A X X i , A Y X i ) − X j g ( T U j X, T U j Y ) i − g ( X, Y ) h ˆ S ( V, W ) − g ( N, T V W ) + X i ( g (( ∇ X i T ) V W, X i )+ g ( A X i V, A X i W )) i(cid:27) ,g ( L ∗ ( U, V ) W, X ) = g (( ∇ U T ) V W, X ) − g (( ∇ V T ) U W, X ) − n − (cid:26) g ( V, W ) h g ( ∇ U N, X ) − X j g ( ∇ U j T ) U j U, X )+ X i { g (( ∇ X i A ) X i X, U ) − g ( A X X i , T U X i ) } i − g ( U, W ) h g ( ∇ V N, X ) − X j g ( ∇ U j T ) U j V, X )+ X i { g (( ∇ X i A ) X i X, V ) − g ( A X X i , T V X i ) } i(cid:27) ew curvature tensors along Riemannian submersion 11 and g ( L ∗ ( U, V ) W, F ) = g ( ˆ R ( U, V ) W, F ) + g ( T U W, T V F ) − g ( T V W, T U F ) − n − (cid:26) g ( V, W ) h ˆ S ( U, F ) − g ( N, T U F )+ X i ( g (( ∇ X i T ) U F, X i ) + g ( A X i U, A X i F )) i − g ( U, W ) h ˆ S ( V, F ) − g ( N, T V F )+ X i ( g (( ∇ X i T ) V F, X i ) + g ( A X i V, A X i F )) i + g ( F, U ) h ˆ S ( U, V ) − g ( N, T U V )+ X i ( g (( ∇ X i T ) U V, X i ) + g ( A X i U, A X i V )) i − g ( F, V ) h ˆ S ( U, W ) − g ( N, T U W )+ X i ( g (( ∇ X i T ) U W, X i ) + g ( A X i U, A X i W )) i(cid:27) . Proof.
Let’s prove the 3 th equation of this theorem. The following equationsare obtained inner production with W to L ∗ and by using equation (1.5) g ( L ∗ ( X, Y ) V, W ) = g ( R M ( X, Y ) V, W ) − n − { g ( X, W ) S ( Y, V ) − g ( Y, W ) S ( X, V ) + g ( Y, V ) S ( X, W ) − g ( X, V ) S ( Y, W ) } . One can easily obtain the other equations by using Theorem 1.2 and Propo-sition 1.4. (cid:3)
Corollary 4.3.
Let π : ( M, g ) → ( G, g ′ ) be a Riemannian submersion, where ( M, g ) and ( G, g ′ ) Riemannian manifolds. If the Riemannian submersion hastotal umbilical fibres, that is N = 0 , then the conharmonic curvature tensor is given by g ( L ∗ ( X, Y ) Z, H )= g ( R G ( X, Y ) Z, H ) + 2 g ( A X Y, A Z H ) − g ( A Y Z, A X H ) + g ( A X Z, A Y H ) − n − ( g ( Y, Z ) (cid:20) S G ( X ′ , H ′ ) ◦ π − X i g ( A X X i , A H X i ) − X j g ( T U j X, T U j H ) (cid:21) − g ( X, Z ) (cid:20) S G ( Y ′ , H ′ ) ◦ π − X i g ( A Y X i , A H X i ) − X j g ( T U j Y, T U j H ) (cid:21) + g ( X, H ) (cid:20) S G ( Y ′ , Z ′ ) ◦ π − X i g ( A Y X i , A Z X i ) − X j g ( T U j Y, T U j Z ) − g ( Y, H ) (cid:20) S G ( X ′ , Z ′ ) ◦ π − X i g ( A X X i , A Z X i ) − X j g ( T U j X, T U j Z ) (cid:21)) ,g ( L ∗ ( X, Y ) Z, V ) = − g (( ∇ Z A ) X Y, V ) − g ( A X T, T V Z )+ g ( A Y Z, T V X ) − g ( A X Z, T V Y ) − n − ( g ( Y, Z ) (cid:20) − X j g (( ∇ U j T ) U j X, V )+ X i ( g (( ∇ X i A ) X i X, V ) − g ( A V X i , T X X i )) (cid:21) − g ( X, Z ) (cid:20) − X j g (( ∇ U j T ) U j Y, V )+ X i ( g (( ∇ X i A ) X i Y, V ) − g ( A V X i , T Y X i )) (cid:21)) ,g ( L ∗ ( X, V ) Y, W )= g (( ∇ X T ) V W, Y ) + g (( ∇ V A ) X Y, W ) − g ( T V X, T W Y ) + g ( A X Y, A Y W ) − n − (cid:26) − g ( V, W ) h S G ( X ′ , Y ′ ) ◦ π − X i g ( A X X i , A Y X i ) − X j g ( T U j X, T U j Y ) i − g ( X, Y ) h ˆ S ( V, W ) + X i ( g (( ∇ X i T ) V W, X i ) + g ( A X i V, A X i W )) i(cid:27) , ew curvature tensors along Riemannian submersion 13 g ( L ∗ ( U, V ) W, X ) = g (( ∇ U T ) V W, X ) − g (( ∇ V T ) U W, X ) − n − (cid:26) g ( V, W ) h X j g ( ∇ U j T ) U j U, X )+ X i { g (( ∇ X i A ) X i X, U ) − g ( A X X i , T U X i ) } i − g ( U, W ) h X j g ( ∇ U j T ) U j V, X )+ X i { g (( ∇ X i A ) X i X, V ) − g ( A X X i , T V X i ) } i(cid:27) and g ( L ∗ ( U, V ) W, F )= g ( ˆ R ( U, V ) W, F ) + g ( T U W, T V F ) − g ( T V W, T U F ) − n − (cid:26) g ( V, W ) h ˆ S ( U, F )+ X i ( g (( ∇ X i T ) U F, X i ) + g ( A X i U, A X i F )) i − g ( U, W ) h ˆ S ( V, F ) + X i ( g (( ∇ X i T ) V F, X i ) + g ( A X i V, A X i F )) i + g ( F, U ) h ˆ S ( U, V ) + X i ( g (( ∇ X i T ) U V, X i ) + g ( A X i U, A X i V )) i − g ( F, V ) h ˆ S ( U, W ) + X i ( g (( ∇ X i T ) U W, X i ) + g ( A X i U, A X i W )) i(cid:27) .
5. Conformal curvature tensor along a Riemannian submersion
In this section, we find some curvature relations of conformal curvature tensorin a Riemannian submersion and give a corollary in case of the Riemanniansubmersion has totally umbilical fibres.
Definition 5.1.
In the n − dimensional space V n , the tensor V ∗ ( X, Y, Z, H ) = R M ( X, Y, Z, H ) − n − g ( X, H ) Ric ( Y, Z ) − g ( Y, H ) Ric ( X, Z )+ g ( Y, Z ) Ric ( X, H ) − g ( X, Z ) Ric ( Y, H )]+ r M ( n − n −
2) [ g ( X, H ) g ( Y, Z ) − g ( Y, H ) g ( X, Z )] , is called conformal curvature tensor, where Ricci tensor and scalar tensordenoted by Ric and r M respectively [10]. Theorem 5.2.
Let ( M, g ) and ( G, g ′ ) Riemannian manifolds, π : ( M, g ) → ( G, g ′ )4 M. A. Akyol and G. Ayar a Riemannian submersion and R M , R G and ˆ R be Riemannian curvaturetensors, S M , S G and ˆ S be Ricci tensors and r M , r G and ˆ r be scalar curvaturetensors of M, G and the fibre respectively. Then for any
U, V, W, F ∈ χ v ( M ) and X, Y, Z, H ∈ χ h ( M ) , we have the following relations g ( V ∗ ( X, Y ) Z, H )= g ( R G ( X, Y ) Z, H ) + 2 g ( A X Y, A Z H ) − g ( A Y Z, A X H ) + g ( A X Z, A Y H ) − n − ( g ( X, H ) (cid:20) S G ( Y ′ , Z ′ ) ◦ π + 12 ( g ( ∇ Y N, Z ) + g ( ∇ Z N, Y )) − X i g ( A Y X i , A Z X i ) − X j g ( T U j Y, T U j Z ) − g ( Y, H ) (cid:20) S G ( X ′ , Z ′ ) ◦ π + 12 ( g ( ∇ X N, Z ) + g ( ∇ Z N, X )) − X i g ( A X X i , A Z X i ) − X j g ( T U j X, T U j Z ) (cid:21) + g ( Y, Z ) (cid:20) S G ( X ′ , H ′ ) ◦ π + 12 ( g ( ∇ X N, H ) + g ( ∇ H N, X )) − X i g ( A X X i , A H X i ) − X j g ( T U j X, T U j H ) (cid:21) − g ( X, Z ) (cid:20) S G ( Y ′ , H ′ ) ◦ π + 12 ( g ( ∇ Y N, H ) + g ( ∇ H N, Y )) − X i g ( A Y X i , A H X i ) − X j g ( T U j Y, T U j H ) (cid:21)) + r M ( n − n − { g ( Y, Z ) g ( X, H ) − g ( X, Z ) g ( Y, H ) } ,g ( V ∗ ( X, Y ) Z, V ) = − g (( ∇ Z A ) X Y, V ) − g ( A X T, T V Z )+ g ( A Y Z, T V X ) − g ( A X Z, T V Y ) − n − ( g ( Y, Z ) (cid:20) g ( ∇ X N, V ) − X j g (( ∇ U j T ) U j X, V )+ X i ( g (( ∇ X i A ) X i X, V ) − g ( A V X i , T X X i )) (cid:21) − g ( X, Z ) (cid:20) g ( ∇ Y N, V ) − X j g (( ∇ U j T ) U j Y, V )+ X i ( g (( ∇ X i A ) X i Y, V ) − g ( A V X i , T Y X i )) (cid:21)) , ew curvature tensors along Riemannian submersion 15 g ( V ∗ ( X, Y ) V, W ) = g (( ∇ V A ) X Y, W ) − g (( ∇ W A ) X Y, V ) + g ( A X V, A Y W ) − g ( A X W, A Y V ) − g ( T V X, T W Y ) + g ( T W X, T V Y ) ,g ( V ∗ ( X, V ) Y, W )= g (( ∇ X T ) V W, Y ) + g (( ∇ V A ) X Y, W ) − g ( T V X, T W Y ) + g ( A X Y, A Y W ) − n − (cid:26) − g ( V, W ) h S G ( X ′ , Y ′ ) ◦ π + 12 ( g ( ∇ X N, Y ) + g ( ∇ Y N, X )) − X i g ( A X X i , A Y X i ) − X j g ( T U j X, T U j Y ) i − g ( X, Y ) h ˆ S ( V, W ) − g ( N, T V W ) + X i ( g (( ∇ X i T ) V W, X i )+ g ( A X i V, A X i W )) i(cid:27) ,g ( V ∗ ( U, V ) W, X ) = g (( ∇ U T ) V W, X ) − g (( ∇ V T ) U W, X ) − n − (cid:26) g ( V, W ) h g ( ∇ U N, X ) − X j g ( ∇ U j T ) U j U, X )+ X i { g (( ∇ X i A ) X i X, U ) − g ( A X X i , T U X i ) } i − g ( U, W ) h g ( ∇ V N, X ) − X j g ( ∇ U j T ) U j V, X )+ X i { g (( ∇ X i A ) X i X, V ) − g ( A X X i , T V X i ) } i(cid:27) and g ( V ∗ ( U, V ) W, F ) = g ( ˆ R ( U, V ) W, F ) + g ( T U W, T V F ) − g ( T V W, T U F ) − n − (cid:26) g ( F, U ) h ˆ S ( V, W ) − g ( N, T V W )+ X i ( g ( ∇ X i T ) V W, X i + g ( A X i V, A X i W )) i − g ( F, V ) h ˆ S ( U, W ) − g ( N, T U W ) + X i ( g (( ∇ X i T ) U W, X i ) + g ( A X i U, A X i W )) i + g ( V, W ) h ˆ S ( U, F ) − g ( N, T U F ) + X i ( g (( ∇ X i T ) U F, X i ) + g ( A X i U, A X i F )) i − g ( U, W ) h ˆ S ( V, F ) − g ( N, T V F ) + X i ( g (( ∇ X i T ) V F, X i ) + g ( A X i V, A X i F )) i(cid:27) + r M ( n − n − { g ( V, W ) g ( U, F ) − g ( U, W ) g ( V, F ) } where r M = ˆ r + r G ◦ π − || N || − || A || − || T || + 2 X i g ( ∇ X i N, X i ) . Proof.
Let’s prove the 4 th equation of this theorem. The following equationsare obtained inner production with W to V ∗ g ( V ∗ ( X, V ) Y, W ) = g ( R M ( X, V ) Y, W ) − n − { g ( X, W ) S M ( V, Y ) − g ( V, W ) S M ( X, Y ) + g ( V, Y ) S ( X, W ) − g ( X, Y ) S M ( V, W ) } + r M ( n − n − { g ( X, W ) g ( Y, V ) − g ( X, V ) g ( Y, W ) } . Then using equations (1.6)-(1.10), we have the desired result. From the The-orem 1.2, Proposition 1.4 and Proposition 1.5 the above equations are ob-tained. (cid:3)
Corollary 5.3.
Let π : ( M, g ) → ( G, g ′ ) be a Riemannian submersion, where ( M, g ) and ( G, g ′ ) Riemannian manifolds. If the Riemannian submersion hastotal umbilical fibres, that is N = 0 , then the conformal curvature tensor isgiven by g ( V ∗ ( X, Y ) Z, H )= g ( R G ( X, Y ) Z, H ) + 2 g ( A X Y, A Z H ) − g ( A Y Z, A X H ) + g ( A X Z, A Y H ) − n − ( g ( X, H ) (cid:20) S G ( Y ′ , Z ′ ) ◦ π − X i g ( A Y X i , A Z X i ) − X j g ( T U j Y, T U j Z ) − g ( Y, H ) (cid:20) S G ( X ′ , Z ′ ) ◦ π − X i g ( A X X i , A Z X i ) − X j g ( T U j X, T U j Z ) (cid:21) + g ( Y, Z ) (cid:20) S G ( X ′ , H ′ ) ◦ π − X i g ( A X X i , A H X i ) − X j g ( T U j X, T U j H ) (cid:21) − g ( X, Z ) (cid:20) S G ( Y ′ , H ′ ) ◦ π − X i g ( A Y X i , A H X i ) − X j g ( T U j Y, T U j H ) (cid:21)) + r M ( n − n − { g ( Y, Z ) g ( X, H ) − g ( X, Z ) g ( Y, H ) } ,g ( V ∗ ( X, Y ) Z, V )= − g (( ∇ Z A ) X Y, V ) − g ( A X T, T V Z ) + g ( A Y Z, T V X ) − g ( A X Z, T V Y ) − n − ( g ( Y, Z ) (cid:20) − X j g (( ∇ U j T ) U j X, V ) + X i ( g (( ∇ X i A ) X i X, V ) − g ( A V X i , T X X i )) (cid:21) − g ( X, Z ) (cid:20) − X j g (( ∇ U j T ) U j Y, V ) + X i ( g (( ∇ X i A ) X i Y, V ) − g ( A V X i , T Y X i )) (cid:21)) , ew curvature tensors along Riemannian submersion 17 g ( V ∗ ( X, V ) Y, W )= g (( ∇ X T ) V W, Y ) + g (( ∇ V A ) X Y, W ) − g ( T V X, T W Y ) + g ( A X Y, A Y W ) − n − (cid:26) − g ( V, W ) h S G ( X ′ , Y ′ ) ◦ π − X i g ( A X X i , A Y X i ) − X j g ( T U j X, T U j Y ) i − g ( X, Y ) h ˆ S ( V, W ) + X i ( g (( ∇ X i T ) V W, X i ) + g ( A X i V, A X i W )) i(cid:27) ,g ( V ∗ ( U, V ) W, X ) = g (( ∇ U T ) V W, X ) − g (( ∇ V T ) U W, X ) − n − (cid:26) g ( V, W ) h X j g ( ∇ U j T ) U j U, X )+ X i { g (( ∇ X i A ) X i X, U ) − g ( A X X i , T U X i ) } i − g ( U, W ) h X j g ( ∇ U j T ) U j V, X )+ X i { g (( ∇ X i A ) X i X, V ) − g ( A X X i , T V X i ) } i(cid:27) and g ( V ∗ ( U, V ) W, F ) = g ( ˆ R ( U, V ) W, F ) + g ( T U W, T V F ) − g ( T V W, T U F ) − n − (cid:26) g ( F, U ) h ˆ S ( V, W )+ X i ( g ( ∇ X i T ) V W, X i + g ( A X i V, A X i W )) i − g ( F, V ) h ˆ S ( U, W ) + X i ( g (( ∇ X i T ) U W, X i ) + g ( A X i U, A X i W )) i + g ( V, W ) h ˆ S ( U, F ) + X i ( g (( ∇ X i T ) U F, X i ) + g ( A X i U, A X i F )) i − g ( U, W ) h ˆ S ( V, F ) + X i ( g (( ∇ X i T ) V F, X i ) + g ( A X i V, A X i F )) i(cid:27) + r M ( n − n − { g ( V, W ) g ( U, F ) − g ( U, W ) g ( V, F ) } where r M = ˆ r + r G ◦ π − || A || − || T || . Finally, we investigate the M − projective curvature tensor on a Rie-mannian submersion and give a corollary in case of the totally umbilicalfibres.8 M. A. Akyol and G. Ayar M -projective curvature tensor along a Riemanniansubmersion In this section, curvature relations of M -projective curvature tensor in a Rie-mannian submersion are examined and obtain a corollary using the curvaturetensor. Definition 6.1.
Let take an n -dimensional differentiable manifold M n withdifferentiability class C ∞ . In 1971 on a n -dimensional Riemannian manifold,ones [13] defined a tensor field W ∗ as W ∗ ( X, Y ) Z = R M ( X, Y ) Z − n −
1) [ S M ( Y, Z ) X − S M ( X, Z ) Y + g ( Y, Z ) QX − g ( X, Z ) QY ]tensor W ∗ as M -projective curvature tensor.In addition, on an n − dimensional Riemannian manifold M n the Riccioperator Q is defined by S M ( X, Y ) = g ( QX, Y ) . Theorem 6.2.
Let ( M, g ) and ( G, g ′ ) Riemannian manifolds, π : ( M, g ) → ( G, g ′ ) a Riemannian submersion and R M , R G and ˆ R be Riemannian curvaturetensors, S M , S G and ˆ S be Ricci tensors of M, G and the fibre respectively.Then for any
U, V, W, F ∈ χ v ( M ) and X, Y, Z, H ∈ χ h ( M ) , we have the ew curvature tensors along Riemannian submersion 19 following relations for M -projective curvature tensor: g ( W ∗ ( X, Y ) Z, H )= g ( R G ( X, Y ) Z, H ) + 2 g ( A X Y, A Z H ) − g ( A Y Z, A X H ) + g ( A X Z, A Y H ) − n − ( g ( X, H ) (cid:20) S G ( Y ′ , Z ′ ) ◦ π + 12 ( g ( ∇ Y N, Z ) + g ( ∇ Z N, Y )) − X i g ( A Y X i , A Z X i ) − X j g ( T U j Y, T U j Z ) − g ( Y, H ) (cid:20) S G ( X ′ , Z ′ ) ◦ π + 12 ( g ( ∇ X N, Z ) + g ( ∇ Z N, X )) − X i g ( A X X i , A Z X i ) − X j g ( T U j X, T U j Z ) (cid:21) + g ( Y, Z ) (cid:20) S G ( X ′ , H ′ ) ◦ π + 12 ( g ( ∇ X N, H ) + g ( ∇ H N, X )) − X i g ( A X X i , A H X i ) − X j g ( T U j X, T U j H ) (cid:21) − g ( X, Z ) (cid:20) S G ( Y ′ , H ′ ) ◦ π + 12 ( g ( ∇ Y N, H ) + g ( ∇ H N, Y )) − X i g ( A Y X i , A H X i ) − X j g ( T U j Y, T U j H ) (cid:21)) ,g ( W ∗ ( X, Y ) Z, V ) = − g (( ∇ Z A ) X Y, V ) − g ( A X Y, T V Z ) + g ( A Y Z, T V X ) − g ( A X Z, T V Y ) − n − ( g ( Y, Z ) (cid:20) g ( ∇ X N, V ) − X j g (( ∇ U j T ) U j X, V )+ X i ( g (( ∇ X i A ) X i X, V ) − g ( A V X i , T X X i )) (cid:21) − g ( X, Z ) (cid:20) g ( ∇ Y N, V ) − X j g (( ∇ U j T ) U j Y, V )+ X i ( g (( ∇ X i A ) X i Y, V ) − g ( A V X i , T Y X i )) (cid:21)) ,g ( W ∗ ( X, Y ) V, W ) = g (( ∇ V A ) X Y, W ) − g (( ∇ W A ) X Y, V ) + g ( A X V, A Y W ) − g ( A X W, A Y V ) − g ( T V X, T W Y ) + g ( T W X, T V Y ) , g ( W ∗ ( X, V ) Y, W ) = g (( ∇ X T ) V W, Y ) + g (( ∇ V A ) X Y, W ) − g ( T V X, T W Y ) + g ( A X Y, A Y W ) − n − (cid:26) − g ( V, W ) h S G ( X ′ , Y ′ ) ◦ π + 12 ( g ( ∇ X N, Y ) + g ( ∇ Y N, X )) − X i g ( A X X i , A Y X i ) − X j g ( T U j X, T U j Y ) i − g ( X, Y ) h ˆ S ( V, W ) − g ( N, T V W )+ X i ( g (( ∇ X i T ) V W, X i ) + g ( A X i V, A X i W )) i(cid:27) ,g ( W ∗ ( U, V ) W, X ) = g (( ∇ U T ) V W, X ) − g (( ∇ V T ) U W, X ) − n − (cid:26) g ( V, W ) h g ( ∇ U N, X ) − X j g ( ∇ U j T ) U j U, X )+ X i { g (( ∇ X i A ) X i X, U ) − g ( A X X i , T U X i ) } i − g ( U, W ) h g ( ∇ V N, X ) − X j g ( ∇ U j T ) U j V, X )+ X i { g (( ∇ X i A ) X i X, V ) − g ( A X X i , T V X i ) } i(cid:27) and g ( W ∗ ( U, V ) W, F ) = g ( ˆ R ( U, V ) W, F ) + g ( T U W, T V F ) − g ( T V W, T U F ) − n − (cid:26) g ( F, U ) h ˆ S ( V, W ) − g ( N, T V W )+ X i ( g ( ∇ X i T ) V W, X i + g ( A X i V, A X i W )) i − g ( F, V ) h ˆ S ( U, W ) − g ( N, T U W ) + X i g (( ∇ X i T ) U W, X i ) + g ( A X i U, A X i W ) i + g ( V, W ) h ˆ S ( U, F ) − g ( N, T U F ) + X i ( g (( ∇ X i T ) U F, X i ) + g ( A X i U, A X i F )) i − g ( U, W ) h ˆ S ( V, F ) − g ( N, T V F ) + X i ( g (( ∇ X i T ) V F, X i ) + g ( A X i V, A X i F )) i(cid:27) . Proof.
Let’s prove the 6 th equation of this theorem. The following equationsare obtained inner production with F to W ∗ and using (1.8) and (1.9) equa-tions. g ( W ∗ ( U, V ) W, F ) = g ( R M ( U, V ) W, F ) − n − (cid:26) g ( F, U ) S M ( U, V ) − g ( F, V ) S M ( U, W ) + g ( V, W ) S M ( U, F ) − g ( U, W ) S M ( V, F ) (cid:27) , ew curvature tensors along Riemannian submersion 21 g ( R M ( U, V ) W, F ) = g ( ˆ R ( U, V ) W, F ) + g ( T U W, T V F ) − g ( T V W, T U F )and S M ( U, V ) = ˆ S ( U, V ) − g ( N, T U V ) + X i { g (( ∇ X i T ) U V, X i ) + g ( A X i U, A X i V ) } . When these equations are substituted in W ∗ , the given result is obtained.Other equations are similarly proved by using Theorem 1.2 and Proposition1.4. (cid:3) Corollary 6.3.
Let π : ( M, g ) → ( G, g ′ ) be a Riemannian submersion, where ( M, g ) and ( G, g ′ ) Riemannian manifolds. If the Riemannian submersion hastotal umbilical fibres, that is N = 0 , then the M -projective curvature tensoris given by g ( W ∗ ( X, Y ) Z, H )= g ( R G ( X, Y ) Z, H ) + 2 g ( A X Y, A Z H ) − g ( A Y Z, A X H ) + g ( A X Z, A Y H ) − n − ( g ( X, H ) (cid:20) S G ( Y ′ , Z ′ ) ◦ π − X i g ( A Y X i , A Z X i ) − X j g ( T U j Y, T U j Z ) (cid:21) − g ( Y, H ) (cid:20) S G ( X ′ , Z ′ ) ◦ π − X i g ( A X X i , A Z X i ) − X j g ( T U j X, T U j Z ) (cid:21) + g ( Y, Z ) (cid:20) S G ( X ′ , H ′ ) ◦ π − X i g ( A X X i , A H X i ) − X j g ( T U j X, T U j H ) (cid:21) − g ( X, Z ) (cid:20) S G ( Y ′ , H ′ ) ◦ π − X i g ( A Y X i , A H X i ) − X j g ( T U j Y, T U j H ) (cid:21)) ,g ( W ∗ ( X, Y ) Z, V ) = − g (( ∇ Z A ) X Y, V ) − g ( A X Y, T V Z ) + g ( A Y Z, T V X ) − g ( A X Z, T V Y ) − n − ( g ( Y, Z ) (cid:20) − X j g (( ∇ U j T ) U j X, V )+ X i ( g (( ∇ X i A ) X i X, V ) − g ( A V X i , T X X i )) (cid:21) − g ( X, Z ) (cid:20) − X j g (( ∇ U j T ) U j Y, V )+ X i ( g (( ∇ X i A ) X i Y, V ) − g ( A V X i , T Y X i )) (cid:21)) , g ( W ∗ ( X, V ) Y, W ) = g (( ∇ X T ) V W, Y )+ g (( ∇ V A ) X Y, W ) − g ( T V X, T W Y )+ g ( A X Y, A Y W ) − n − (cid:26) − g ( V, W ) h S G ( X ′ , Y ′ ) ◦ π − X i g ( A X X i , A Y X i ) − X j g ( T U j X, T U j Y ) i − g ( X, Y ) h ˆ S ( V, W ) + X i ( g (( ∇ X i T ) V W, X i ) + g ( A X i V, A X i W )) i(cid:27) ,g ( W ∗ ( U, V ) W, X ) = g (( ∇ U T ) V W, X ) − g (( ∇ V T ) U W, X ) − n − (cid:26) g ( V, W ) h X j g ( ∇ U j T ) U j U, X )+ X i { g (( ∇ X i A ) X i X, U ) − g ( A X X i , T U X i ) } i − g ( U, W ) h X j g ( ∇ U j T ) U j V, X )+ X i { g (( ∇ X i A ) X i X, V ) − g ( A X X i , T V X i ) } i(cid:27) and g ( W ∗ ( U, V ) W, F )= g ( ˆ R ( U, V ) W, F ) + g ( T U W, T V F ) − g ( T V W, T U F ) − n − (cid:26) g ( F, U ) h ˆ S ( V, W ) + X i ( g ( ∇ X i T ) V W, X i + g ( A X i V, A X i W )) i − g ( F, V ) h ˆ S ( U, W ) + X i g (( ∇ X i T ) U W, X i ) + g ( A X i U, A X i W ) i + g ( V, W ) h ˆ S ( U, F ) + X i ( g (( ∇ X i T ) U F, X i ) + g ( A X i U, A X i F )) i − g ( U, W ) h ˆ S ( V, F ) + X i ( g (( ∇ X i T ) V F, X i ) + g ( A X i V, A X i F )) i(cid:27) . Remark 6.4.
The authors investigate new curvature tensors along Riemann-ian submersions and obtain some results by using totally umbilical fibres.Therefore, it will be worth examining new curvature tensors along Riemann-ian submersions.
Acknowledgement.
Both authors would like to thank Prof. Dr. Gabriel Ed-uard Vilcu for some useful comments and questions that help to clarify somestatements of the original manuscript.
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Mehmet Akif AkyolBingol UniversityFaculty of Arts and Sciences,Department of Mathematics12000, Bing¨ol, Turkeye-mail: [email protected]