Slant Riemannian Submersions from Sasakian Manifolds
aa r X i v : . [ m a t h . DG ] S e p SLANT RIEMANNIAN SUBMERSIONS FROM SASAKIANMANIFOLDS
I. K ¨UPELI ERKEN AND C. MURATHAN
Abstract.
We introduce slant Riemannian submersions from Sasakian manifoldsonto Riemannian manifolds. We survey main results of slant Riemannian submersionsdefined on Sasakian manifolds. We also give an example of such slant submersions. Introduction
Let F be a C ∞ -submersion from a Riemannian manifold ( M, g M ) onto a Riemannianmanifold ( N, g N ) . Then according to the conditions on the map F : ( M, g M ) → ( N, g N ) , we have the following submersions:semi-Riemannian submersion and Lorentzian submersion [11], Riemannian submer-sion ([19], [12]), slant submersion ([9], [24]), almost Hermitian submersion [27], contact-complex submersion [16], quaternionic submersion [15], almost h -slant submersion and h -slant submersion [21], semi-invariant submersion [25], h -semi-invariant submersion [22],etc. As we know, Riemannian submersions are related with physics and have their appli-cations in the Yang-Mills theory ([6], [28]), Kaluza-Klein theory ([7], [13]), Supergravityand superstring theories ([14], [29]). In [23], Sahin introduced anti-invariant Riemann-ian submersions from almost Hermitian manifolds onto Riemannian manifolds. He alsosuggested to investigate anti invariant submersions from almost contact metric manifoldsonto Riemannian manifolds in [26].So the purpose of the present paper is to study similar problems for slant Riemanniansubmersions from Sasakian manifolds to Riemannian manifolds. We also want to carryanti-invariant submanifolds of Sasakian manifolds to anti-invariant Riemannian submer-sion theory and to prove dual results for submersions. For instance, a slant submanifoldof a K -contact manifold is an anti invariant submanifold if and only if ∇ Q = 0 (see:Proposition 4.1 of [8]). We get similar result as Proposition 4. Thus, it will be worththe study area which is anti-invariant submersions from almost contact metric manifoldsonto Riemannian manifolds.The paper is organized as follows: In section 2, we present the basic information aboutRiemannian submersions needed for this paper. In section 3, we mention about Sasakianmanifolds. In section 4, we give definition of slant Riemannian submersions and introduceslant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. Wesurvey main results of slant submersions defined on Sasakian manifolds. We also give anexample of slant submersions such that characteristic vector field ξ is vertical. Date : 10.09.2013.2000
Mathematics Subject Classification.
Primary 53C25, 53C43, 53C55; Secondary 53D15.
Key words and phrases.
Riemannian submersion, Sasakian manifold, Anti-invariant submersionThis paper is supported by Uludag University research project (KUAP(F)-2012/57). Riemannian Submersions
In this section we recall several notions and results which will be needed throughoutthe paper.Let (
M, g M ) be an m -dimensional Riemannian manifold , let ( N, g N ) be an n -dimensionalRiemannian manifold. A Riemannian submersion is a smooth map F : M → N which isonto and satisfies the following axioms: S F has maximal rank. S
2. The differential F ∗ preserves the lenghts of horizontal vectors.The fundamental tensors of a submersion were defined by O’Neill ([19],[20]). They are(1 , M , given by the formula: T ( E, F ) = T E F = H∇ V E V F + V∇ V E H F, (2.1) A ( E, F ) = A E F = V∇ H E H F + H∇ H E V F, (2.2)for any vector field E and F on M. Here ∇ denotes the Levi-Civita connection of ( M, g M ).These tensors are called integrability tensors for the Riemannian submersions. Note thatwe denote the projection morphism on the distributions ker F ∗ and (ker F ∗ ) ⊥ by V and H , respectively. The following Lemmas are well known ([19],[20]). Lemma 1.
For any
U, W vertical and
X, Y horizontal vector fields, the tensor fields T , A satisfy: i ) T U W = T W U, (2.3) ii ) A X Y = −A Y X = 12 V [ X, Y ] . (2.4)It is easy to see that T is vertical, T E = T V E and A is horizontal, A = A H E .For each q ∈ N, F − ( q ) is an ( m − n ) dimensional submanifold of M . The submanifolds F − ( q ) , q ∈ N, are called fibers. A vector field on M is called vertical if it is alwaystangent to fibers. A vector field on M is called horizontal if it is always orthogonal tofibers. A vector field X on M is called basic if X is horizontal and F -related to a vectorfield X on N, i. e., F ∗ X p = X ∗ F ( p ) for all p ∈ M. Lemma 2.
Let F : ( M, g M ) → ( N, g N ) be a Riemannian submersion. If X, Y are basicvector fields on M , then: i ) g M ( X, Y ) = g N ( X ∗ , Y ∗ ) ◦ F,ii ) H [ X, Y ] is basic, F -related to [ X ∗ , Y ∗ ], iii ) H ( ∇ X Y ) is basic vector field corresponding to ∇ ∗ X ∗ Y ∗ where ∇ ∗ is the connectionon N.iv ) for any vertical vector field V , [ X, V ] is vertical.Moreover, if X is basic and U is vertical then H ( ∇ U X ) = H ( ∇ X U ) = A X U. On theother hand, from (2.1) and (2.2) we have ∇ V W = T V W + ˆ ∇ V W (2.5) ∇ V X = H∇ V X + T V X (2.6) ∇ X V = A X V + V∇ X V (2.7) ∇ X Y = H∇ X Y + A X Y (2.8)for X, Y ∈ Γ((ker F ∗ ) ⊥ ) and V, W ∈ Γ(ker F ∗ ) , where ˆ ∇ V W = V∇ V W. Notice that T acts on the fibres as the second fundamental form of the submersionand restricted to vertical vector fields and it can be easily seen that T = 0 is equivalent LANT RIEMANNIAN SUBMERSIONS FROM SASAKIAN MANIFOLDS 3 to the condition that the fibres are totally geodesic. A Riemannian submersion is calleda Riemannian submersion with totally geodesic fiber if T vanishes identically. Let U , ..., U m − n be an orthonormal frame of Γ(ker F ∗ ) . Then the horizontal vector field H = m − n m − n X j =1 T U j U j is called the mean curvature vector field of the fiber. If H = 0 theRiemannian submersion is said to be minimal. A Riemannian submersion is called aRiemannian submersion with totally umbilical fibers if(2.9) T U W = g M ( U, W ) H for U, W ∈ Γ(ker F ∗ ). For any E ∈ Γ( T M ) , T E and A E are skew-symmetric operatorson (Γ( T M ) , g M ) reversing the horizontal and the vertical distributions. By Lemma 1horizontally distribution H is integrable if and only if A =0. For any D, E, G ∈ Γ( T M )one has(2.10) g ( T D E, G ) + g ( T D G, E ) = 0 , (2.11) g ( A D E, G ) + g ( A D G, E ) = 0 . We recall the notion of harmonic maps between Riemannian manifolds. Let (
M, g M )and ( N, g N ) be Riemannian manifolds and suppose that ϕ : M → N is a smoothmap between them. Then the differential ϕ ∗ of ϕ can be viewed a section of the bun-dle Hom ( T M, ϕ − T N ) → M, where ϕ − T N is the pullback bundle which has fibres( ϕ − T N ) p = T ϕ ( p ) N, p ∈ M. Hom ( T M, ϕ − T N ) has a connection ∇ induced from theLevi-Civita connection ∇ M and the pullback connection. Then the second fundamentalform of ϕ is given by(2.12) ( ∇ ϕ ∗ )( X, Y ) = ∇ ϕX ϕ ∗ ( Y ) − ϕ ∗ ( ∇ MX Y )for X, Y ∈ Γ( T M ) , where ∇ ϕ is the pullback connection. It is known that the secondfundamental form is symmetric. If ϕ is a Riemannian submersion it can be easily provethat(2.13) ( ∇ ϕ ∗ )( X, Y ) = 0for
X, Y ∈ Γ((ker F ∗ ) ⊥ ).A smooth map ϕ : ( M, g M ) → ( N, g N ) is said to be harmonicif trace ( ∇ ϕ ∗ ) = 0 . On the other hand, the tension field of ϕ is the section τ ( ϕ ) ofΓ( ϕ − T N ) defined by(2.14) τ ( ϕ ) = divϕ ∗ = m X i =1 ( ∇ ϕ ∗ )( e i , e i ) , where { e , ..., e m } is the orthonormal frame on M . Then it follows that ϕ is harmonic ifand only if τ ( ϕ ) = 0, for details, [2].3. Sasakian Manifolds A n -dimensional differentiable manifold M is said to have an almost contact structure( φ, ξ, η ) if it carries a tensor field φ of type (1 , ξ and 1-form η on M respectively such that(3.1) φ = − I + η ⊗ ξ, φξ = 0 , η ◦ φ = 0 , η ( ξ ) = 1 , where I denotes the identity tensor. I. K¨UPELI ERKEN AND C. MURATHAN
The almost contact structure is said to be normal if N + dη ⊗ ξ = 0, where N is theNijenhuis tensor of φ . Suppose that a Riemannian metric tensor g is given in M andsatisfies the condition(3.2) g ( φX, φY ) = g ( X, Y ) − η ( X ) η ( Y ) , η ( X ) = g ( X, ξ ) . Then ( φ, ξ, η, g )-structure is called an almost contact metric structure. Define a tensorfield Φ of type (0 ,
2) by Φ(
X, Y ) = g ( φX, Y ). If dη = Φ then an almost contact metricstructure is said to be normal contact metric structure. A normal contact metric structureis called a Sasakian structure, which satisfies(3.3) ( ∇ X φ ) Y = g ( X, Y ) ξ − η ( Y ) X, where ∇ denotes the Levi-Civita connection of g . For a Sasakian manifold M = M n +1 ,it is known that(3.4) R ( ξ, X ) Y = g ( X, Y ) ξ − η ( Y ) X, (3.5) S ( X, ξ ) = 2 nη ( X )and(3.6) ∇ X ξ = − φX. [5].Now we will introduce a well known Sasakian manifold example on R n +1 . Example 1 ([4]) . We consider R n +1 with Cartesian coordinates ( x i , y i , z ) ( i = 1 , ..., n ) and its usual contact form η = 12 ( dz − n X i =1 y i dx i ) . The characteristic vector field ξ is given by ∂∂z and its Riemannian metric g and tensorfield φ are given by g = 14 ( η ⊗ η + n X i =1 (( dx i ) + ( dy i ) ) , φ = δ ij − δ ij y j , i = 1 , ..., n This gives a contact metric structure on R n +1 . The vector fields E i = 2 ∂∂y i , E n + i =2 (cid:16) ∂∂x i + y i ∂∂z (cid:17) , ξ form a φ -basis for the contact metric structure. On the other hand, itcan be shown that R n +1 ( φ, ξ, η, g ) is a Sasakian manifold. Slant Riemannian submersionsDefinition 1.
Let M ( φ, ξ, η, g M ) be a Sasakian manifold and ( N, g N ) be a Riemannianmanifold. A Riemannian submersion F : M ( φ, ξ, η, g M ) → ( N, g N ) is said to be slant iffor any non zero vector X ∈ Γ(ker F ∗ ) − { ξ } , the angle θ ( X ) between φX and the space ker F ∗ is a constant (which is independent of the choice of p ∈ M and of X ∈ Γ(ker F ∗ ) −{ ξ } ). The angle θ is called the slant angle of the slant submersion. Invariant and anti-invariant submersions are slant submersions with θ = 0 and θ = π/ , respectively. Aslant submersion which is not invariant nor anti-invariant is called proper submersion. Now we will introduce an example.
LANT RIEMANNIAN SUBMERSIONS FROM SASAKIAN MANIFOLDS 5
Example 2. R has got a Sasakian structure as in Example 1. Let F : R → R bea map defined by F ( x , x , y , y , z ) = ( x − √ x + y , x − √ x + y ) . Then, bydirect calculations ker F ∗ = span { V = 2 E + 1 √ E , V = E , V = ξ = E } and (ker F ∗ ) ⊥ = span { H = 2 E − √ E , H = E } . Then it is easy to see that F is a Riemannian submersion. Moreover, φV = 2 E − √ E and φV = E imply that | g ( φV , V ) | = √ . So F is a slant submersion with slant angle θ = π . In Example 2, we note that the characteristic vector field ξ is vertical. If ξ is orthogonalto ker F ∗ we will give following Theorem. Theorem 1.
Let F be a slant Riemannian submersion from a Sasakian manifold M ( φ, ξ, η, g M ) onto a Riemannian manifold ( N, g N ) . If ξ is orthogonal to ker F ∗ , then F is anti-invariant.Proof. By (3.6), (2.6), (2.10) and (2.3) we have g ( φU, V ) = − g ( ∇ U ξ, V ) = − g ( T U ξ, V ) = g ( T U V, ξ )= g ( T V U, ξ ) = g ( U, φV )for any
U, V ∈ Γ(ker F ∗ ) . Using skew symmetry property of φ in the last relation wecomplete the proof of the Theorem. (cid:3) Remark 1.
We note Lotta [17] proved that if M is a submanifold of contact metricmanifold of ˜ M and ξ is orthogonal to M , then M is anti-invariant submanifold. So,our result can be seen as a submersion version of Lotta’s result Now, let F be a slant Riemannian submersion from a Sasakian manifold M ( φ, ξ, η, g M )onto a Riemannian manifold ( N, g N ) . Then for any
U, V ∈ Γ(ker F ∗ ) , we put(4.1) φU = ψU + ωU, where ψU and ωU are vertical and horizontal components of φU , respectively. Similarly,for any X ∈ Γ(ker F ∗ ) ⊥ , we have(4.2) φX = B X + C X, where B X (resp. C X ) , is vertical part (resp. horizontal part) of φX .From (3.2), (4.1) and (4.2) we obtain(4.3) g M ( ψU, V ) = − g M ( U, ψV )and(4.4) g M ( ωU, Y ) = − g M ( U, B Y ) . for any U, V ∈ Γ(ker F ∗ ) and Y ∈ Γ((ker F ∗ ) ⊥ ).Using (2.5), (4.1) and (3.6) we obtain(4.5) T U ξ = − ωU, ˆ ∇ U ξ = − ψU for any U ∈ Γ(ker F ∗ ) . I. K¨UPELI ERKEN AND C. MURATHAN
Now we will give the following proposition for a Riemannian submersion with twodimensional fibers which is similar to Proposition 3.2. of [1].
Proposition 1.
Let F be a Riemannian submersion from almost contact manifold onto aRiemannian manifold. If dim( ker F ∗ ) = 2 and ξ is vertical then fibers are anti-invariant. As the proof of the following proposition is similar to slant submanifolds (see [8]) wedon’t give its proof.
Proposition 2.
Let F be a Riemannian submersion from a Sasakian manifold M ( φ, ξ, η, g M ) onto a Riemannian manifold ( N, g N ) such that ξ ∈ Γ(ker F ∗ ) . Then F is anti-invariantsubmersion if and only if D is integrable, where D = ker F ∗ − { ξ } . Theorem 2.
Let M ( φ, ξ, η, g M ) be a Sasakian manifold of dimension m +1 and ( N, g N ) is a Riemannian manifold of dimension n . Let F : M ( φ, ξ, η, g M ) → ( N, g N ) be a slantRiemannian submersion. Then the fibers are not totally umbilical.Proof. Using (2.5) and (3.6) we obtain(4.6) T U ξ = − ωU for any U ∈ Γ(ker F ∗ ). If the fibers are totally umbilical, then we have T U V = g M ( U, V ) H for any vertical vector fields U, V where H is the mean curvature vector field of any fibre.Since T ξ ξ = 0, we have H = 0, which shows that fibres are minimal. Hence the fibersare totally geodesic, which is a contradiction to the fact that T U ξ = − ωU = 0. (cid:3) By (2.5), (2.6), (4.1) and (4.2) we have(4.7) ( ∇ U ω ) V = C T U V − T U ψV, (4.8) ( ∇ U φ ) V = BT U V − T U ωV + R ( ξ, U ) V, where ( ∇ U ω ) V = H∇ U ωV − ω ˆ ∇ U V ( ∇ U ψ ) V = ˆ ∇ U ψV − ψ ˆ ∇ U V, for U, V ∈ Γ(ker F ∗ ) . Now we will characterize slant submersion by following Theorem.
Theorem 3.
Let F be a Riemannian submersion from a Sasakian manifold M ( φ, ξ, η, g M ) onto a Riemannian manifold ( N, g N ) such that ξ ∈ Γ(ker F ∗ ) . Then, F is a slant Rie-mannian submersion if and only if there exist a constant λ ∈ [0 , such that (4.9) ψ = − λ ( I − η ⊗ ξ ) . Furthermore, in such case, if θ is the slant angle of F , it satisfies that λ = cos θ. Proof.
Firstly we suppose that F is not an anti-invariant Riemannian submersion. Then,for U ∈ Γ(ker F ∗ ) , (4.10) cos θ = g M ( φU, ψU ) | φU | | ψU | = | ψU | | φU | | ψU | = | ψU || φU | . Since φU ⊥ ξ, we have g ( ψU, ξ ) = 0 . Now, substituting U by ψU in (4.10) and using(3.2) we obtain(4.11) cos θ = (cid:12)(cid:12) ψ U (cid:12)(cid:12) | φψU | = (cid:12)(cid:12) ψ U (cid:12)(cid:12) | ψU | . LANT RIEMANNIAN SUBMERSIONS FROM SASAKIAN MANIFOLDS 7
From (4.10) and (4.11) we have(4.12) | ψU | = (cid:12)(cid:12) ψ U (cid:12)(cid:12) | φU | On the other hand, one can get following g M ( ψ U, U ) = g M ( φψU, U ) = − g M ( ψU, φU )(4.13) = − g M ( ψU, ψU ) = − | ψU | . Using (4.12) and (4.13) we get g M ( ψ U, U ) = − (cid:12)(cid:12) ψ U (cid:12)(cid:12) | φU | = − (cid:12)(cid:12) ψ U (cid:12)(cid:12) (cid:12)(cid:12) φ U (cid:12)(cid:12) (4.14)Also, one can easily get(4.15) g M ( ψ U, φ U ) = − g M ( ψ U, U ) . So, by help (4.14) and (4.15) we obtain g M ( ψ U, φ U ) = (cid:12)(cid:12) ψ U (cid:12)(cid:12) (cid:12)(cid:12) φ U (cid:12)(cid:12) and it follows that ψ U and φ U are colineal, that is ψ U = λφ U = − λ ( I − η ⊗ ξ ) . Using the last relationtogether with (4.10) and (4.11) we obtain cos θ = √ λ is constant and so F is a slantRiemannian submersion.If F is anti-invariant Riemannian submersion φU is normal, ψU = 0 and it is equiv-alent to ψ U = 0 . In this case θ = π and so the equation (4.10) is again provided. (cid:3) By using (3.2), (4.1), (4.3) and (4.9) we have following Lemma.
Lemma 3.
Let F be a slant Riemannian submersion from a Sasakian manifold M ( φ, ξ, η, g M ) onto a Riemannian manifold ( N, g N ) with slant angle θ. Then the following relations arevalid (4.16) g M ( ψU, ψV ) = cos θ ( g M ( U, V ) − η ( U ) η ( V )) , (4.17) g M ( ωU, ωV ) = sin θ ( g M ( U, V ) − η ( U ) η ( V )) for any U, V ∈ Γ(ker F ∗ ) . We denote the complementary orthogonal distribution to ω (ker F ∗ ) in (ker F ∗ ) ⊥ by µ. Then we have(4.18) (ker F ∗ ) ⊥ = ω (ker F ∗ ) ⊕ µ. Lemma 4.
Let F be a proper slant Riemannian submersion from a Sasakian manifold M ( φ, ξ, η, g M ) onto a Riemannian manifold ( N, g N ) then µ is an invariant distributionof (ker F ∗ ) ⊥ , under the endomorphism φ .Proof. For X ∈ Γ( µ ) , from (3.2) and (4.1), we obtain g M ( φX, ωV ) = g M ( φX, φV ) − g M ( φX, ψV )= g M ( X, V ) − η ( X ) η ( V ) − g M ( φX, ψV )= − g M ( X, φψV ) . Using (4.9) and (4.18) we have g M ( φX, ωV ) = − cos θg M ( X, V − η ( V ) ξ )= g M ( X, ωψV )= 0 . I. K¨UPELI ERKEN AND C. MURATHAN
In a similar way, we have g M ( φX, U ) = − g M ( X, φU ) = 0 due to φU ∈ Γ((ker F ∗ ) ⊕ ω (ker F ∗ )) for X ∈ Γ( µ ) and U ∈ Γ(ker F ∗ ) . Thus the proof of the lemma is completed. (cid:3)
By help (4.17), we can give following
Corollary 1.
Let F be a proper slant Riemannian submersion from a Sasakian manifold M m +1 ( φ, ξ, η, g M ) onto a Riemannian manifold ( N n , g N ) . Let { e , e , ...e m − n , ξ } be a local orthonormal basis of (ker F ∗ ) , then { csc θwe , csc θwe , ..., csc θwe m − n } is alocal orthonormal basis of ω (ker F ∗ ) . By using (4.18) and Corollary 1 one can easily prove the following Proposition.
Proposition 3.
Let F be a proper slant Riemannian submersion from a Sasakian mani-fold M m +1 ( φ, ξ, η, g M ) onto a Riemannian manifold ( N n , g N ) . Then dim ( µ ) = 2( n − m ) . If µ = { } , then n = m. By (4.3) and (4.16) we have
Lemma 5.
Let F be a proper slant Riemannian submersion from a Sasakian manifold M m +1 ( φ, ξ, η, g M ) onto a Riemannian manifold ( N n , g N ) . If e , e , ...e k , ξ are orthogonalunit vector fields in (ker F ∗ ) , then { e , sec θψe , e , sec θψe , ...e k , sec θψe k , ξ } is a local orthonormal basis of (ker F ∗ ) . Moreover dim (ker F ∗ ) = 2 m − n + 1 = 2 k + 1 and dim N = n = 2( m − k ) . Lemma 6.
Let F be a slant Riemannian submersion from a Sasakian manifold M ( φ, ξ, η, g M ) onto a Riemannian manifold ( N, g N ) If ω is parallel then we have (4.19) T ψU ψU = − cos θ ( T U U + η ( U ) ωU ) Proof. If ω is parallel, from (4.7), we obtain C T U V = T U ψV for U, V ∈ Γ(ker F ∗ ) . Weinterchange U and V and use (2.3) we get T U ψV = T V ψU. Substituting V by ψU in the above equation and then using Theorem 3 we get therequired formula. (cid:3) We give a sufficient condition for a slant Riemannian submersion to be harmonic as ananalogue of a slant Riemannian submersion from a Sasakian manifold onto a Riemannianmanifold in [24].
Theorem 4.
Let F be a slant Riemannian submersion from a Sasakian manifold M ( φ, ξ, η, g M ) onto a Riemannian manifold ( N, g N ) If ω is parallel then F is a harmonic map.Proof. From [10] we know that F is harmonic if and only if F has minimal fibres. Thus F is harmonic if and only if n X i =1 T e i e i = 0 . Thus using the adapted frame for slantRiemannian submersion and by the help of (2.14) and Lemma 5 we can write τ = − m − n X i =1 F ∗ ( T e i e i + T sec θψe i sec θψe i ) − F ∗ ( T ξ ξ ) . LANT RIEMANNIAN SUBMERSIONS FROM SASAKIAN MANIFOLDS 9
Using T ξ ξ = 0 we have τ = − m − n X i =1 F ∗ ( T e i e i + sec θ T ψe i ψe i )By virtue of (4.19) in the above equation, we obtain τ = − m − n X i =1 F ∗ ( T e i e i + sec θ ( − cos θ ( T e i e i + η ( e i ) ωe i )))= − m − n X i =1 F ∗ ( T e i e i − T e i e i ) = 0So we prove that F is harmonic. (cid:3) Now setting Q = ψ ,we define ∇ Q by( ∇ U Q ) V = V∇ U QV − Q ˆ ∇ U V for any U, V ∈ Γ(ker F ∗ ) . We give a characterization for a slant Riemannian submersionfrom a Sasakian manifold M ( φ, ξ, η, g M ) onto a Riemannian manifold ( N, g N ) by usingthe value of ∇ Q . Proposition 4.
Let F be a slant Riemannian submersion from a Sasakian manifold M ( φ, ξ, η, g M ) onto a Riemannian manifold ( N, g N ) . Then, ∇ Q = 0 if and only if F isan anti-invariant submersion.Proof. By using (4.9),(4.20) Q ˆ ∇ U V = − cos θ ( ˆ ∇ U V − η ( ˆ ∇ U V ) ξ )for each U, V ∈ Γ(ker F ∗ ) , where θ is slant angle.On the other hand,(4.21) V ( ∇ U QV ) = − cos θ ( ˆ ∇ U V − η ( ˆ ∇ U V ) ξ + g ( V, ψU ) ξ + η ( V ) ψU ) . So, from (4.20) and ∇ Q = 0 if and only if cos θ ( g ( V, ψU ) ξ + η ( V ) ψU ) = 0 which impliesthat ψU = 0 or θ = π . Both the cases verify that F is an anti-invariant submersion. (cid:3) We now investigate the geometry of leaves of (ker F ∗ ) ⊥ and ker F ∗ . Proposition 5.
Let F be a slant Riemannian submersion from a Sasakian manifold M ( φ, ξ, η, g M ) onto a Riemannian manifold ( N, g N ) . Then the distribution (ker F ∗ ) ⊥ de-fines a totally geodesic foliation on M if and only if g M ( H∇ X Y, ωψU ) − sin θg M ( Y, φX ) η ( U ) = g M ( A X B Y, ωU ) + g M ( H∇ X C Y, ωU ) for any X, Y ∈ Γ((ker F ∗ ) ⊥ ) and U ∈ Γ(ker F ∗ ) . Proof.
From (3.3) and (4.1)we have g M ( ∇ X Y, U ) = − g M ( φ ∇ X φY, U ) + g M ( Y, φX ) η ( U )(4.22) = g M ( ∇ X φY, φU ) + g M ( Y, φX ) η ( U )= g M ( ∇ X φY, ψU ) + g M ( ∇ X φY, ωU ) + g M ( Y, φX ) η ( U ) . for any X, Y ∈ Γ((ker F ∗ ) ⊥ ) and U ∈ Γ(ker F ∗ ). Using (3.3) and (4.1) in (4.22), we obtain g M ( ∇ X Y, U ) = − g M ( ∇ X Y, ψ U ) − g M ( ∇ X Y, ωψU )(4.23) + g M ( Y, φX ) η ( U ) + g M ( ∇ X φY, ωU ) . By (4.2) and (4.9) we have g M ( ∇ X Y, U ) = cos θg M ( ∇ X Y, U ) − cos θη ( U ) η ( ∇ X Y )(4.24) − g M ( ∇ X Y, ωψU ) + g M ( Y, φX ) η ( U )+ g M ( ∇ X B Y, ωU ) + g M ( ∇ X C Y, ωU )Using (2.7), (2.8) and (3.6) in the last equation we obtainsin θg M ( ∇ X Y, U ) = sin θg M ( Y, φX ) η ( U ) − g M ( H∇ X Y, ωψU )+ g M ( A X B Y, ωU ) + g M ( H∇ X C Y, ωU )which prove the theorem. (cid:3)
Proposition 6.
Let F be a slant Riemannian submersion from a Sasakian manifold M ( φ, ξ, η, g M ) onto a Riemannian manifold ( N, g N ) . If the distribution ker F ∗ defines atotally geodesic foliation on M then F is an invariant submersion.Proof. By (4.5), if the distribution ker F ∗ defines a totally geodesic foliation on M thenwe conclude that ωU = 0 for any U ∈ Γ(ker F ∗ ) which shows that F is an invariantsubmersion. (cid:3) OpenProblem:
Let F be a slant Riemannian submersion from a Sasakian manifold M ( φ, ξ, η, g M )onto a Riemannian manifold ( N, g N ). In [3],Barrera et.al. define and study the Maslovform of non-invariant slant submanifolds of S -space form ˜ M ( c ). They find conditions forit to be closed. By similar discussion in [3] we can define Maslov form Ω H of M as thedual form of the vector field B H , that is,Ω H ( U ) = g M ( U, B H )for any U ∈ Γ(ker F ∗ ) . So.it will be interesting for giving a chararacterization respectto Ω H for slant submersions, where H = m − n X i =1 T e i e i + T sec θψe i sec θψe i and { e , sec θψe , e , sec θψe , ...e k , sec θψe k , ξ } is a local orthonormal basis of (ker F ∗ ) . References [1]
Alegre P.,
Slant submanifolds of Lorentzian Saakian and Para Sasakian manifolds,
Tai-wanese Journal of Mathematics., 17, No. 2, (2012), 629-659. [2]
Baird P., Wood J.C. , Harmonic Morphisms Between Riemannian Manifolds,
London Math-ematical Society Monographs, 29, Oxford University Press, The Clarendon Press, Oxford,2003. [3]
Barrera j., Carriazo A., Fernandez L.M., Prieto-Martin A.
The Maslow form in non-invariant slant submanifolds of S-space-forms,
Ann. Mat. Pura Appl. 4, (2012), 803-818. [4]
Blair D.E.,
Contact manifolds in Riemannian geometry , Lectures Notes in Mathematics , Springer-Verlag, Berlin, (1976), 146p [5]
Blair D. E.,
Riemannian geometry of contact and symplectic manifolds,
Progress in Math-ematics. 203, Birkhauser Boston, Basel, Berlin, 2002. [6]
Bourguignon J. P. , Lawson H. B. ,
Stability and isolation phenomena for Yang-mills fields,
Commum. Math. Phys. 79, (1981), 189-230. [7]
Bourguignon J. P. , Lawson H. B. ,
A Mathematician’s visit to Kaluza-Klein theory,
Rend.Semin. Mat. Torino Fasc. Spec. (1989), 143-163.
LANT RIEMANNIAN SUBMERSIONS FROM SASAKIAN MANIFOLDS 11 [8]
Cabrerizo J. L., Carriazo A., Fernandez L. M. and Fernandez M.,
Slant submanifolds inSasakian Manifolds,
Glasgow. Math. 42, (2000), 125-138. [9]
Chen B. Y. ,
Geometry of slant submanifolds,
Katholieke Universiteit Leuven, Leuven, 1990. [10]
Eells J., Sampson J. H.,
Harmonic Mappings of Riemannian Manifolds,
Amer. J. Math.,86, (1964), 109-160. [11]
Falcitelli M. , Ianus S. , and Pastore A. M. , Riemannian submersions and related topics, , World Scientific Publishing Co., 2004. [12]
Gray A. ,
Pseudo-Riemannian almost product manifolds and submersions,
J. Math. Mech,16 (1967), 715-737. [13]
IanusS. , Visinescu M. ,
Kaluza-Klein theory with scalar fields and generalized Hopf mani-folds,
Class. Quantum Gravity 4, (1987), 1317-1325. [14]
IanusS. , Visinescu M.
Space-time compactification and Riemannian submersions,
In: Ras-sias, G.(ed.) The Mathematical Heritage of C. F. Gauss, (1991), 358-371, World Scientific,River Edge. [15]
Ianus S. , Mazzocco R. , Vilcu G. E. ,
Riemannian submersions from quaternionic manifolds , Acta. Appl. Math. 104, (2008) 83-89. [16]
Ianus S. , Ionescu A. M. , Mazzocco R. , Vilcu G. E. ,
Riemannian submersions from almostcontact metric manifolds, arXiv: 1102.1570v1 [math. DG]. [17]
Lotta A.,
Slant submanifolds in contact geometry,
Bull. Math. Soc. Roumanie 39, (1996),183-198. [18]
Yano K., Kon Masahiro. ,
Anti-invariant submanifolds ,
Marcel Dekker, Inc. New York andBassel, 1976. [19]
O’Neill B.,
The fundamental equations of submersion,
Michigan Math. J. 13, (1966) 459-469. [20]
O’Neill B.,
Semi-Riemannian geometry with applications to relativity,
Academic Press, NewYork-London 1983. [21]
Park K. S. ,
H–slant submersions , Bull. Korean Math. Soc. 49 No. 2, (2012), 329-338. [22]
Park K. S. ,
H-semi-invariant submersions,
Taiwan. J. Math., 16(5), (2012), 1865-1878. [23]
Sahin B. ,
Anti-invariant Riemannian submersions from almost Hermitian manifolds , Cent.Eur. J. Math. 8(3), (2010) 437-447. [24]
Sahin B. ,
Slant submersions from almost Hermitian manifolds , Bull. Math. Soc. Sci. Math.Roumanie Tome 54(102) No. 1, (2011), 93 - 105. [25]
Sahin B. ,
Semi-invariant submersions from almost Hermitian manifolds , Canad. Math.Bull. , 54, No. 3 (2011) [26] S ahin B. ,
Riemannian submersions from almost Hermitian manifolds , Taiwanese Journalof Mathematics., 17, No. 2, (2012), 629-659. [27]
Watson B. ,
Almost Hermitian submersions,
J. Differential Geom. , 11(1), (1976), 147-165. [28]
Watson B. ,
G, G ′ - Riemannian submersions and nonlinear gauge field equations of generalrelativity,
In: Rassias, T. (ed.) Global Analysis - Analysis on manifolds, dedicated M. Morse.Teubner-Texte Math., 57 (1983), 324-349, Teubner, Leipzig. [29]
M. T. Mustafa,
Applications of harmonic morphisms to gravity,
J. Math. Phys. , 41(10),(2000), 6918-6929.
Art and Science Faculty,Department of Mathematics, Uludag University, 16059 Bursa,TURKEY
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