POINTWISE SLANT SUBMERSIONS FROM KENMOTSU MANIFOLDS INTO RIEMANNIAN MANIFOLDS

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1 ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N ( ) 561 POINTWISE SLANT SUBMERSIONS FROM KENMOTSU MANIFOLDS INTO RIEMANNIAN MANIFOLDS Sushil Kumar Department of Mathematics Astronomy University of Lucknow Lucknow (U.P.)-India Amit Kumar Rai University School of Basic Applied Scinces Guru Gobind singh Indraprastha University New Delhi-India Rajendra Prasad Department of Mathematics Astronomy University of Lucknow Lucknow (U.P.)-India Abstract. The purpose of this paper is to study pointwise slant submersions from Kenmotsu manifolds onto Riemannian manifolds admitting vertical horizontal structure vector fields find some results related to totally geodesic harmonic properties. Keywords: Riemannian submersion, almost contact manifold, pointwise slant submersion 1. Introduction Immersions submersions are special tools in Differential Geometry. Both play important role in Riemannian Geometry. O Neill [19] Gray [14] introduced Riemannian submersions between Riemannian manifolds. Submersions between Riemannian manifolds equipped with an additional structure of almost contact type on total space, firstly studied by Watson [23] Chinea [11] independently. We know that Riemannian submersions are related to Mathematical Physics have their applications in the Kaluza-Klein theory ([7], [16]) the Yang-Mills theory [6] etc. On the other h, submersions have been studied by several authors. Some related research papers are: Geometry of slant submanifolds [10], Slant submersions from almost Hermitian manifolds [20], Slant submanifolds of Lorentzian Sasakian para Sasakian manifolds [1], Riemannian submersions from almost. Corresponding author

2 SUSHIL KUMAR, AMIT KUMAR RAI, RAJENDRA PRASAD 562 contact metric manifolds [15], On quasi-slant submanifolds of an almost Hermitian manifold [12], Slant submanifolds in Sasakian manifolds [8], Pointwise slant submanifolds in almost Hermitian manifolds [9], Almost contact metric submersions [11], Riemannian Submersions Related Topics [13], Pointwise slant submersions [17], Slant submanifolds of a Riemannian product manifold [2], Slant submanifolds in contact geometry [18], Point-wise slant submanifolds in almost contact geometry [3], Pointwise slant submersions from Cosymplectic manifolds [21] etc. In this paper, we study pointwise slant submersions from Kenmotsu manifolds onto Riemannian manifolds. The paper is organized as follows. In section 2, we collect main notions formulae which are needed for this paper. In section 3, we obtain some results of pointwise slant submersions from Kenmotsu manifolds onto Riemannian manifolds admitting vertical horizontal structure vector fields. 2. Preliminaries Let M be an almost contact metric manifold. So there exist on M, a (1, 1) tensor field ϕ, a vector field ξ, a 1 form η a Riemannian metric g such that (2.1) (2.2) ϕ 2 = I + η ξ, ϕ ξ = 0, η ϕ = 0, g(x, ξ) = η(x), η(ξ) = 1, (2.3) g(ϕx, ϕy ) = g(x, Y ) η(x)η(y ), g(ϕx, Y ) = g(x, ϕy ), for any vector fields X Y on M I is the identity tensor field [5]. An almost contact metric manifold M equipped with an almost contact metric structure (ϕ, ξ, η, g) is denoted by (M, ϕ, ξ, η, g). An almost contact metric manifold M is called a Kenmotsu manifold if (2.4) ( X ϕ)y = g(ϕx, Y )ξ η(y )ϕx, for any vector fields X Y on M, where is the Riemannian connection of the Riemannian metric g. If (M, ϕ, ξ, η, g) be a Kenmotsu manifold, then the following equation holds: (2.5) X ξ = X η(x)ξ. Let M be an m dimensional Riemannian manifold N be an n dimensional Riemannian manifold (m > n) with Riemannian metrics g M g N respectively. Let f : (M, g M ) (N, g N ) be a C map. We denote the kernel space of f by ker f consider the orthogonal complementary space H = (ker f ) to ker f. Then the tangent bundle of M has the following decomposition (2.6) T M = (ker f ) (ker f ).

3 POINTWISE SLANT SUBMERSIONS We also denote the range of f by rangef consider the orthogonal complementary space (rangef ) to rangef in the tangent bundle T N of N. Thus the tangent bundle T N of N has the following decomposition (2.7) T N = (rangef ) (rangef ). A Riemannian submersion f is a C map from Riemannian manifold (M, g M ) onto (N, g N ) satisfying the following conditions: (i) f has the maximal rank, (ii) The differential f preserves the lengths of horizontal vectors. For each x N, f 1 (x) is fiber which is a (m n) dimensional submanifold of M. If a vector field on M is always tangent (resp. orthogonal) to fibers, then it is called vertical (resp. horizontal). A vector field X on M is said to basic if it is horizontal f related to a vector field X on N, i.e., f X p = X f(p) for all p M. We denote the projection morphisms on the distributions ker f (ker f ) by V H respectively. A smooth map f : (M, g M ) (N, g N ) between Riemannian manifolds is a Riemannian submersion if only if (2.8) g M (U, V ) = g N (f U, f V ), for every U, V (ker f ). The O Neill s tensors T A define by (2.9) (2.10) T E F = H VE VF + V VE HF, A E F = V HE HF + H HE VF, for arbitrary vector fields E F on M, where is the Riemannian connection on M [19]. Lemma 1. Let f be a Riemannian submersion between Riemannian manifolds (M, g M ) (N, g N ). If X Y are basic vector fields on M, then: (i) g M (X, Y ) = g N (f X, f Y ), (ii) the horizontal part [X, Y ] H of [X, Y ] is a basic vector field corresponds to [X, Y ] i.e., f ([X, Y ] H ) = [X, Y ], (iii) [V, X] is vertical for any vector field V of ker f, (iv) ( M X Y )H is vertical for any vector field corresponding to N X Y where M N are the Riemannian connection on M N respectively. Now, from equations (2.9) (2.10), we get (2.11) (2.12) (2.13) (2.14) X Y = T X Y + X Y, X V = H X V + T X V, V X = A V X + V V X, V W = H V W + A V W,

4 SUSHIL KUMAR, AMIT KUMAR RAI, RAJENDRA PRASAD 564 for X, Y Γ(ker f ) V, W Γ(ker f ), where X Y = V X Y. Moreover, if V is basic, then H X V = A V X. On the other h, for any E Γ(T M), it is seen that T is vertical, T E = T VE A is horizontal, A E = A HE. The tensor fields T A satisfy the equations: (2.15) (2.16) T X Y = T Y X, A V W = A W V = 1 V[V, W ], 2 for X, Y Γ(ker F ) V, W Γ(ker f ). It can be easily seen that a Riemannian submersion f : (M, g M ) (N, g N ) has totally geodesic fibers if only if T identically vanishes. Now, we consider the notion of harmonic maps between Riemannian manifolds. Let (M, g M ) (N, g N ) be Riemannian manifolds suppose that f is a C mapping between them. Then the differential f of f can be considered as a section of the bundle Hom(T M, f 1 T N) M, where f 1 T N is the pullback bundle that has fibers (f 1 T N) q = T f(q) N, q M. If Hom(T M, f 1 T N) has a connection induced from the Riemannian connection M, then the second fundamental form of f is given by (2.17) ( f )(X, Y ) = f X f Y f ( M X Y ), for any X, Y Γ(T M), where f is the pullback connection. If f is a Riemannian submersion, then we can easily see that (2.18) ( f )(V, W ) = 0, for any V, W Γ(ker f ) [4]. 3. The pointwise slant submersions from almost contact metric manifolds Let f be a Riemannian submersion from a Kenmotsu manifold (M, ϕ, ξ, η, g M ) onto a Riemannian manifold (N, g N ). If for each x M, the angle θ(x) between ϕx the space ker f is independent of the choice of the non-zero vector field X Γ(ker f ) {ξ}, then f is called a pointwise slant submersion the angle θ is said to be slant function of the pointwise slant submersion. A pointwise slant submersion is called slant if its slant function θ is independent of the choice of the point on (M, ϕ, ξ, η, g M ). Then the constant θ is called the slant angle of the slant submersion [21]. 3.1 Pointwise slant submersion for ξ Γ(ker f ) Let f be a Riemannian submersion from a Kenmotsu manifold (M, ϕ, ξ, η, g M ) onto a Riemannian manifold (N, g N ).

5 POINTWISE SLANT SUBMERSIONS For any X Γ(ker f ), we get (3.1) ϕx = ψx + ωx, where ψx ωx are vertical horizontal components of ϕx respectively. For any V Γ(ker f ), we have (3.2) ϕv = BV + CV, where BV CV are vertical horizontal components of ϕv respectively By using equations (2.3), (3.1) (3.2), we get (3.3) (3.4) g M (ψx, Y ) = g M (X, ψy ), g M (ωx, V ) = g M (X, BV ), for any X, Y Γ(ker f ) V Γ(ker f ). Again using the equations (2.5), (2.11), (2.13), (3.1) (3.2), we get (3.5) X ξ = X η(x)ξ, T X ξ = 0, for any X Γ(ker f ) V Γ(ker f ). For any X, Y Γ(ker f ), define (3.6) ( X ψ)y = X ψy ψ X Y, (3.7) ( X ω)y = H X ωy ω X Y, where is the Riemannian connection on M. Next, we say that the ω is parallel if (3.8) ( X ω)y = 0. Then we easily have: Lemma 2. Let (M, ϕ, ξ, η, g M ) be a Kenmotsu manifold (N, g N ) be a Riemannian manifold. If f : (M, ϕ, ξ, η, g M ) (N, g N ) is a pointwise slant submersion, then (3.9) ( X ψ)y = BT X Y T X ωy g(ψx, Y )ξ + η(y )ψx, (3.10) ( X ω)y = CT X Y T X ψy + η(y )ωx, for any X, Y Γ(ker f ).

6 SUSHIL KUMAR, AMIT KUMAR RAI, RAJENDRA PRASAD 566 In the same way with the proof of Theorem 1 in [21], we have the following theorem: Theorem 1. If (M, ϕ, ξ, η, g M ) be a Kenmotsu manifold (N, g N ) be a Riemannian manifold. If f : (M, ϕ, ξ, η, g M ) (N, g N ) is a pointwise slant submersion, then ψ 2 = cos 2 θ( I + η ξ). Simlarly, as in the proof of Lemma 2 in [21], we have Corollary 1. Let (M, ϕ, ξ, η, g M ) be a Kenmotsu manifold (N, g N ) be a Riemannian manifold. If f : (M, ϕ, ξ, η, g M ) (N, g N ) is a pointwise slant submersion, then we have for any X, Y Γ(ker f ). g M (ψx, ψy ) = cos 2 θ(g M (X, Y ) η(x)η(y )), g M (ωx, ωy ) = sin 2 θ(g M (X, Y ) η(x)η(y )), In the same way with the proof of Lemma 3 in [21], we can state the following theorem: Theorem 2. Let (M, ϕ, ξ, η, g M ) be a Kenmotsu manifold (N, g N ) be a Riemannian manifold. Assume that f : (M, ϕ, ξ, η, g M ) (N, g N ) is a pointwise slant submersion. If ω is parallel, then we have for any X Γ(ker f ). T ψx ψx = cos 2 θ(t X X η(x)ωψx), Let f be a C map from Riemannian manifold (M, g M ) onto (N, g N ), then the adjoint f map of f is characterized by (3.11) g M (X, f q Y ) = g N (f q X, Y ), for any X T q M, Y T f(q) N q M. For each q M, f h defined by is a C map f h : ((ker f ) (q), g M (ker f ) (q)) (rangef (q), g N (rangef )(q)), where denote the adjoint of f h by f h. Let f q be the adjoint of f q that is defined by f q : (T q M, g M ) (T f(q) N, g N ). The linear transformation ( f ) h : rangef (q) (ker f ) (q), defined as ( f ) h Y = f Y, where Y Γ(rangef ), is an isomorphism (f q) h 1 = ( f q ) h = (f q). h Theorem 3. Let (M, ϕ, ξ, η, g M ) be a Kenmotsu manifold (N, g N ) be a Riemannian manifold. If f : (M, ϕ, ξ, η, g M ) (N, g N ) is a pointwise slant

7 POINTWISE SLANT SUBMERSIONS submersion with non-zero slant function θ, then the fibers are totally geodesic submanifolds of M if only if g N ( N V f (ωx), f (ωy )) = g M ([X, V ], Y ) sin 2 θ + V (θ)g M (ϕx, ϕy ) sin 2θ + g M (A V ωψx, Y ) g M (A V ωx, ψy ) η(y )g M (BV, ψx) η( V X)η(Y ) sin 2 θ, for any X, Y Γ(ker f ) V Γ(ker f ), where V V are f related vector fields N is the Riemannian connection on N. Proof. For any X, Y Γ(ker f ) V Γ(ker f ), using equations (2.3), (2.4), (2.11) (3.1), we get g M (T X Y, V ) = g M ([X, V ], Y ) + g M ( V ψ 2 X, Y ) + g M ( V ωψx, Y ) g M ( V ωx, ϕy ) η( V X)η(Y ). From theorem 1 using equations (2.11), (2.14) (3.2), we get g M (T X Y, V ) sin 2 θ = g M ([X, V ], Y ) sin 2 θ + V (θ)g M (ϕx, ϕy ) sin 2θ + g M (A V ωψx, Y ) g N ( N V f (ωx), f (ωy )) g M (A V ωx, ψy ) η( V X)η(Y )) sin 2 θ η(y )g M (BV, ψx). By considering the fibers as totally geodesic, we derive the formula in the above theorem. Conversely, it can be directly verified. Theorem 4. Let (M, ϕ, ξ, η, g M ) be a Kenmotsu manifold (N, g N ) be a Riemannian manifold. If f : (M, ϕ, ξ, η, g M ) (N, g N ) be a pointwise slant submersion with non-zero slant function θ, then f is a totally geodesic map if only if g N ( N V f (ωx), f (ωy )) = g M ([X, V ], Y ) sin 2 θ + V (θ)g M (ϕx, ϕy ) sin 2θ + g M (A V ωψx, Y ) g M (A V ωx, ψy ) η(y )g M (BV, ψx) η( V X)η(Y ) sin 2 θ, g M (A V ωx, BW ) = g N ( f V f (ωψx), f (W )) g N ( f V f (ωx), f (CW )) η(x)g M (V, W ) sin 2 θ, for X, Y Γ(ker f ) V, W Γ(ker f ), where V V are f related vector fields f is the pullback connection along f.

8 SUSHIL KUMAR, AMIT KUMAR RAI, RAJENDRA PRASAD 568 Proof. By definition, it follows that f is totally geodesic if only if ( f )(X, Y ) = 0, for any X, Y Γ(T M). From theorem 3, we obtain the first equation. On the other h, for X, Y Γ(ker f ) V, W Γ(ker f ), using equations (2.3), (2.4) (3.1) we get g M ( V X, W ) = g M ( V ψ 2 X, W ) g M ( V ωψx, W ) + g M ( V ωx, ϕw ) + η(x)g M (V, W ). From theorem 1 using equations (2.8), (2.11), (2.14), (2.17) (3.2), we get Converse is obvious. g N (( f )(V, X), f (W )) sin 2 θ = g N ( f V f (ωx), f (W )) + g N ( f V f (ωx), f (CW )) + g M (A V ωx, BW ) + η(x)g M (V, W ) sin 2 θ. Theorem 5. Let (M, ϕ, ξ, η, g M ) be a Kenmotsu manifold (N, g N ) be a Riemannian manifold. If f : (M, ϕ, ξ, η, g M ) (N, g N ) be a pointwise slant submersion with non-zero slant function θ, then f is harmonic if only if trace f (( f )((.)ωψ(.))) tracet (.) ω(.) + tracec f ( f )((.)ω(.)) = 0. Proof. For any X Γ(ker f ) V Γ(ker f ), using equations (2.1), (2.3), (2.11), (3.1) (3.2), we get g M (T X X, V ) = g M (ϕ X ψx, V ) + g M ( X ωx, ϕv ). From theorem 1 using equations (2.3), (2.4) (3.1), we get g M (T X X, V ) = g M ( X X, V ) cos 2 θ g M ( X ωψx, V ) + g M ( X ωx, ϕv ). Using equations (2.12), (2.17), (3.2) (3.11), we have g M (T X X, V ) sin 2 θ = g N ( f ( f )(X, ωψx), V ) g M (ωt X ωx, V ) g N (C f ( f )(X, ωx), V ). Conversely, a direct computation gives the proof. 3.2 Pointwise slant submersions for ξ Γ((ker f ) ) In this section, we give the basic equations of pointwise slant submersions from Kenmotsu manifolds onto Riemannian manifolds for ξ Γ(ker f ). From equations (2.1) (2.2), we get (3.12) ϕ 2 X = X,

9 POINTWISE SLANT SUBMERSIONS (3.13) g(ϕx, ϕy ) = g(x, Y ), for any X, Y Γ(ker f ). Moreover, from equations (2.12), (2.14), (2.5), (3.1) (3.2), we get (3.14) (3.15) T X ξ = X, A V ξ = 0, (3.16) η( X Y ) = g M (X, Y ), for any X, Y Γ(ker f ) V Γ(ker f ). Theorem 6. If (M, ϕ, ξ, η, g M ) be a Kenmotsu manifold (N, g N ) be a Riemannian manifold. If f : (M, ϕ, ξ, η, g M ) (N, g N ) is a pointwise slant submersion, then ψ 2 = (cos 2 θ)i. Corollary 2. Let (M, ϕ, ξ, η, g M ) be a Kenmotsu manifold (N, g N ) be a Riemannian manifold. If f : (M, ϕ, ξ, η, g M ) (N, g N ) be a pointwise slant submersion, then g M (ψx, ψy ) = cos 2 θg M (X, Y ), for any X, Y Γ(ker f ). g M (ωx, ωy ) = sin 2 θ(g M (X, Y ), Theorem 7. Let (M, ϕ, ξ, η, g M ) be a Kenmotsu manifold (N, g N ) be a Riemannian manifold. Assume that f : (M, ϕ, ξ, η, g M ) (N, g N ) is a pointwise slant submersion with slant function θ. If ω is parallel, then for any X Γ(ker f ). T ψx ψx = cos 2 θt X X, Theorem 8. Let (M, ϕ, ξ, η, g M ) be a Kenmotsu manifold (N, g N ) be a Riemannian manifold. If f : (M, ϕ, ξ, η, g M ) (N, g N ) is a pointwise slant submersion with non-zero slant function θ, then the fibers are totally geodesic submanifolds of M if only if g N ( N V f (ωx), f (ωy )) = g M ([X, V ], Y ) sin 2 θ + V (θ)g M (X, Y ) sin 2θ + g M (A V ωψx, Y ) g M (A V ωx, ψy ), for any X, Y Γ(ker f ) V Γ(ker f ), where V V are f related vector fields N is the Riemannian connection on N.

10 SUSHIL KUMAR, AMIT KUMAR RAI, RAJENDRA PRASAD 570 Proof. For any X, Y Γ(ker f ) V Γ(ker f ), using equations (2.2), (2.3), (2.4), (2.11), (2.14), (3.1), (3.2), (2.14) Theorem 6, we get g M (T X Y, V ) sin 2 θ = g M ([X, V ], Y ) sin 2 θ + V (θ)g M (X, Y ) sin 2θ + g M (A V ωψx, Y ) g N ( N V / f (ωx), f (ωy )) g M (A V ωx, ψy ). By considering the fibers as totally geodesic, we derive the formula. Conversely, it can be directly verified. Theorem 9. Let (M, ϕ, ξ, η, g M ) be a Kenmotsu manifold (N, g N ) be a Riemannian manifold. Let f : (M, ϕ, ξ, η, g M ) (N, g N ) be a pointwise slant submersion with non-zero slant function θ, then f is totally geodesic map if only if g N ( N V f (ωx), f (ωy )) = g M ([X, V ], Y ) sin 2 θ + V (θ)g M (X, Y ) sin 2θ + g M (A V ωψx, Y ) g M (A V ωx, ψy ), g M (A V ωx, BW ) = g N ( f V f (ωψx), f (W )) g N ( f V f (ωx), f (CW )) η(w )g M (BV, ψx), for any X, Y Γ(ker f ) V, W Γ(ker f ), where V V are f related vector fields f is the pullback connection along f. Proof. By definition, it follows that f is totally geodesic if only if ( f )(X, Y ) = 0, for any X, Y Γ(T M). From Theorem 3, we obtain the first equation. On the other h, for X, Y Γ(ker f ) V, W Γ(ker f ), using equations (2.2), (2.3), (2.4), (2.10), (2.11), (3.1), (3.2), (2.14), Theorem 6, we obtain g N (( f )(V, X), f (W )) sin 2 θ = g N ( f V f (ωx), f (W )) + g N ( f V f (ωx), f (CW )) + g M (A V ωx, BW ) + η(w )g M (BV, ψx). Conversely, it can be easily proved. References [1] P. Alegre, Slant submanifolds of Lorentzian Sasakian para Sasakian manifolds, Taiwanese J. Math., 17(3) (2013), [2] M. Atceken, Slant submanifolds of a Riemannian product manifold, Acta Math. Sci., 30 B (1) ( 2010),

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