Invariant submanifolds of special trans-sasakian structure

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1 Global Journal of Pure and Applied Mathematics. ISSN Volume 13, Number 4 (017), pp Research India Publications Invariant submanifolds of special trans-sasakian structure Shivaprasanna G.S. Department of Mathematics, Dr. Ambedkar Institute of Technology, Bengaluru , India. Bhavya K. Department of Mathematics, Presidency University, Bengaluru , India. Somashekhara G. Department of Mathematics, M.S. Ramaiah University of Applied Sciences, Bengaluru , India. Abstract The object of present paper is to find necessary and sufficient conditions for invariant submanifolds of special trans-sasakian structure to be totally geodesic. AMS subject classification: Keywords: (ε, δ)trans-sasakian manifold, second fundamental form, invariant submanifold, totally geodesic, semi parallel. 1. Introduction Invariant submanifolds of a contact manifold have been a major area of research for long time since the concept was borrowed from complex geometry. A submanifold of a contact manifold is said to be totally geodesic if every geodesic in that submanifold is also geodesic in the ambient manifold. There is well known result of Kon that an

2 1184 Shivaprasanna G.S., Bhavya K. and Somashekhara G. invariant submanifold of a Sasakian manifold is totally geodesic, provided the second fundamental form of the immersion is covariantly constant [11]. The concept of (ε)-sasakian manifolds was introduced by A.Bejancu and K.L.Duggal [1] and further investigation was taken up by Xufend and Xiaoli[16] and Rakesh kumar et al.[1]. De and Sarkar [6] introduced and studied conformally flat,weyl semisymmetric, φ-recurrent (ε)-kenmotsu manifolds. In [1], the authors obtained Riemannian curvature tensor of (ε)-sasakian manifolds and established relations among different curvatures. H.G.Nagraja et al.[9] introduced (ε, δ)-trans-sasakian structures which generalizes both (ε)-sasakian manifolds and (ε)-kenmotsu manifolds and this structure is called special trans-sasakian structure(sts). In this paper we studied invariant submanifolds of 3- dimensional (ε, δ)-trans-sasakian structure [(ST S) 3 ].. Preliminaries Let (M,g) be an almost contact metric manifold of dimension (n+1) equipped with an almost contact metric structure (φ,ξ,η,g)consisting of a (1,1) tensor field φ, a vector field ξ, a 1-form η and a Riemannian metric g satisfying if φ = I + η ξ, (.1) η(ξ) = 1, (.) φξ = 0,η φ = 0. (.3) An almost contact metric manifold M is called an (ε)-almost contact metric manifold g(ξ,ξ) = ε, (.4) η(x) = εg(x, ξ), (.5) g(φx, φy) = g(x, Y) εη(x)η(y ), X, Y TM, (.6) where ε = g(ξ,ξ) =±1. An almost contact metric structure (φ,ξ,η,g)on a connected manifold M is called (ε, δ)-trans-sasakian manifold structure. If (M R, J, G) belongs to the class W 4 [8], where J is the almost complex structure on M R defined by J(X,fd/dt) = (φx f ξ, η(x)d/dt) (.7) for all vector fields X on M and smooth functions f on M R and G is the product metric on M R.This may be expressed by the condition [3] ( X φ)y = α[g(x, Y)ξ εη(y )X]+β[g(φX, Y)ξ δη(y )φx], (.8) holds for some smooth functions α and β on M and ε =±1, δ =±1. For β = 0,α = 1, an (ε, δ)-trans-sasakian manifold reduces to an (ε)-sasakian and for α = 0,β = 1it reduces to a (δ)-kenmotsu manifold. Then from (.8), it is easy to see that ( X ξ) = εαφx βδφ X, (.9)

3 Invariant submanifolds of special trans-sasakian structure 1185 ( X η)y = αg(y, φx) + εδβg(φx, φy ). (.10) In a (n+1) dimensional (ε, δ)-trans-sasakian manifold (ST S) we also have the following [9], [13]: h(x, ξ) = 0. (.11) R(X, Y)ξ = ε((y α)φx (Xα)φY ) + (β α )(η(x)y η(y)x) δ((xβ)φ Y (Y β)φ X) + εδαβ(η(y )φx η(x)φy) + αβ(δ ε)g(φx, Y )ξ. (.1) S(X,ξ) = (φx)α + ((n 1)(εα β δ) (ξβ))η(x) (n 1)(Xβ), (.13) where S is the Ricci tensor of type (0,) and R curvature tensor of type (1,3). For constants α and β,wehave R(X, Y)ξ = (β α )[η(x)y η(y)x] (.14) ξα + αβδ = 0. (.15) In a 3-dimensionl (ε, δ)-trans-sasakian manifold [(ST S) 3 ], the curvature tensor R and Ricci tensor S are given by [10],[18] R(X, Y)Z =(A r )(g(y, Z)X g(x, Z)Y) + B(g(Y, Z)η(X) g(x, Z)η(Y))ξ + Bη(Z)(η(Y )X η(x)y), (.16) where εδ = 1, A = curvature. From (.16), we have S(X,Y) = R(X, Y)ξ =[(A r )ε + B][η(Y)X η(x)y] (.17) ( r (α β )), B = ( 3(α β ) ε r ) and r is the scalar [ ( A r ) ] + Bε g(x, Y) + B(3 ε)η(x)η(y). (.18) 3. submanifolds of an almost contact metric manifold Let M be a submanifold of a contact manifold M. We denote and the Levi-Civita connections of M and M respectively, and T (M) the normal bundle of M. Then Gauss and Weingarten formulas are given by X Y = X Y + h(x, Y ) (3.1) X N = X N A NX (3.)

4 1186 Shivaprasanna G.S., Bhavya K. and Somashekhara G. for any X, Y TM. is the connection in the normal bundle, h is the second fundamental form of M and A N is the Weingarten endomorphism associated with N. The second fundamental form h and the shape operator A related by From (3.1) we have g(h(x, Y ), N) = g(a N X, Y ). (3.3) X ξ = X ξ + h(x, ξ). (3.4) Let M and M be two Riemannian or semi-riemannian manifolds, f : M M be an immersion, h be the second fundamental form and be the vander-warden-bortolotti connection of M. An immersion is said to be semiparallel if R(X, Y) h = X Y Y X [X,Y ] (3.5) In [] Arslan et.al defined and studied submanifolds satisfying the conditions R(X, Y) h = 0 (3.6) for all vector fields X,Y tangent to M and such manifolds are called -semiparallel. The 3-dimensional Weyl-projective curvature tensor P is defined as By virtue of (.17) and (3.7) P(ξ,Y)Z = B P(X,Y)Z = R(X, Y)Z 1 [g(y, Z)QX g(x, Z)QY] (3.7) [ 1 ε ] g(y, Z)ξ + Bξ P(ξ,Y)ξ = B { (1 ε)η(y)η(z) [ (ε 1)η(Y )ξ ] (3 ε) ξ (3 ε) } [η(y) + 1] 4. Some basic properties of invariant submanifolds of special trans-sasakian structure (3.8) (3.9) Definition 4.1. A submanifold M of a (ST S) 3 -manifold M is said to be invariant if the structure vector field ξ is tangent to M at every point of M and φx is tangent to M for any vector field X tangent to M at every point of M, that is φ(t N) TN at every point of M. The submanifold M of (ST S) 3 -manifold M is called totally geodesic if h(x, Y ) = 0 for any X, Y Ɣ(T N). For the second fundamental form h, the covariant derivative of h is defined by ( X h)(y, Z) = X (h(y, Z)) h( XY, Z) h(y, X Z) (4.1)

5 Invariant submanifolds of special trans-sasakian structure 1187 for any vector fields X, Y, Z tangent to M. Then h is a normal bundle valued tensor of type (0,3) and is called the third fundamental form of M, is called the Vander- Waerden-Bortolotti connection of M. i.e., is the connection in TN T N built with and. If h = 0, then M is to have parallel second fundamental form [17]. From the Gauss and Weingarten formulae we obtain R(X, Y)Z = R(X, Y)Z + A h(x,z) Y A h(y,z) X, (4.) where R(X, Y)Z denotes the tangential part of the curvature tensor of the submanifold. From (3.5), we get (R(X, Y)Z h)(z, U) = R (X, Y )h(z, U) h(r(x, Y )Z, U) h(z, R(X, Y )U) (4.3) for all vector fields X, Y, Z and U, where R (X, Y ) =[ X, Y ] [X,Y ] (4.4) and R denotes the curvature tensor of. In the similar manner we can write (R(X, Y) h)(z, U, W ) =R (X, Y )( h)(z, U, W ) ( h)(r(x, Y )Z, U, W ) ( h)(z, R(X, Y )U, W ) ( h)(z, U, R(X, Y )W ) (4.5) for all fields X, Y, Z, U and W tangent to M and ( h)(z, U, W) = Z h)(u, W). Again for Weyl-projective curvature tensor P we have (P(X,Y) h)(z, U) = R (X, Y )h(z, U) h(p (X, Y )Z, U) h(z, P (X, Y )U) (4.6) For the second fundamental form h, the covariant derivative of h is defined by ( X h)(y, Z) = X (h(y, Z)) h( XY, Z) h(y, X Z) (4.7) for any vector fields X, Y, Z tangent to M. Then h is a normal bundle valued tensor of type (0,3) and is called the third fundamental form of M, is called the Vander- Waerden-Bortolotti connection of M. ie., is the connection in TN T N built with and. If h = 0, then M is to have parallel second fundamental form[17]. From the Gauss and Weingarten formulae we obtain R(X, Y)Z = R(X, Y)Z + A h(x,z) Y A h(y,z) X, (4.8) where R(X, Y)Z denotes the tangential part of the curvature tensor of the submanifold. From (3.5), we get (R(X, Y)Z h)(z, U) = R (X, Y )h(z, U) h(r(x, Y )Z, U) h(z, R(X, Y )U) (4.9)

6 1188 Shivaprasanna G.S., Bhavya K. and Somashekhara G. for all vector fields X, Y, Z and U, where R (X, Y ) =[ X, Y ] [X,Y ] (4.10) and R denotes the curvature tensor of. In the similar manner we can write (R(X, Y) h)(z, U, W ) =R (X, Y )( h)(z, U, W ) ( h)(r(x, Y )Z, U, W ) ( h)(z, R(X, Y )U, W ) ( h)(z, U, R(X, Y )W ) (4.11) for all fields X, Y, Z, U and W tangent to M and ( h)(z, U, W) = Z h)(u, W). Again for Weyl-projective curvature tensor P we have (P(X,Y) h)(z, U) = R (X, Y )h(z, U) h(p (X, Y )Z, U) h(z, P (X, Y )U) (4.1) and (P(X,Y) h)(z, U, W ) =R (X, Y )( h)(z, U, W) ( h)(p (X, Y )Z, U, W ) ( h)(z, P (X, Y )U, W ) ( h)(z, U, P (X, Y )W ) (4.13) 5. Invariant submanifolds of (ST S) 3 -manifolds satisfying Q(h, R) = 0 Theorem 5.1. In an Invariant submanifolds of (ST S) 3 -manifolds satisfying Q(h, R) = 0 with α = β if and only if it is totaly geodesic. Proof. Let an Invariant submanifolds of (ST S) 3 -manifolds satisfying Q(h, R) = 0, Therefore, 0 = Q(h, R)(X, Y, Z; U,V ) = ((U h V, R)(X, Y )Z) = R((U h V)X,Y)Z R(X, (U h V)Y)Z R(X, Y )(U h V)Z, where (U V)is defined by Using (5.) in (5.1), we have (5.1) (U V)W = h(v, W)U h(u, W)V (5.) h(v, X)R(U, Y )Z + h(u, X)R(, Y )Z h(v, Y )R(X, U)Z + h(u, Y )R(X, V )Z h(v, Z)R(X, V )U + h(u, Z)R(X, Y )V = 0. Putting Z = ξ in (5.3) h(v, X)R(U, Y )ξ + h(u, X)R(V, Y )ξ h(v, Y )R(X, U)ξ + h(u, Y )R(X, V )ξ h(v, ξ)r(x, V )U + h(u, ξ)r(x, Y )V = 0. (5.3) (5.4)

7 Invariant submanifolds of special trans-sasakian structure 1189 Again put V = ξ in (5.4), we have h(u, X)R(ξ, Y )ξ + h(u, Y )R(X, ξ)ξ = 0. (5.5) This implies that (3 ε) {h(u, X)[η(Y)η(W) g(y, W)]+h(U, Y )[g(x, W) η(x)η(w)]}. (5.6) Contracting Y and W,weget h(u, X) = 0, provided α = β, (5.7) hence the proof. 6. Invariant submanifolds of (ST S) 3 -manifolds satisfying Q(S, h) = 0 Theorem 6.1. In an Invariant submanifolds of (ST S) 3 -manifolds satisfying Q(S, h) = 0 with B = (r 4A) if and only if totally geodesic. Proof. Let an invariant submanifolds of (ST S) 3 -manifolds satisfying Q(h, R) = 0, therefore, 0 = Q(S, h)(x, Y : U,V) = h(u S V)X,Y) h(x, (U S V)Y), (6.1) where h(u S V)W = S(V,W)U S(U,W)V. (6.) Using (6.) in (6.1), we have S(V, X)h(U, Y )+S(U, X)h(V, Y ) S(V, Y )h(x, U)+S(U, Y )h(x, V ) = 0 (6.3) Putting U = Y = ξ in (6.3), we obtain S(ξ, ξ)h(x, V ) = 0. (6.4) It implies that Hence the proof. h(x, V ) = 0, provide B = (r 4A). (6.5)

8 1190 Shivaprasanna G.S., Bhavya K. and Somashekhara G. 7. -semiparallel invariant submanifolds of (ST S) 3 -manifolds Theorem 7.1. Let M be an invariant submanifold of a (ST S) 3 manifold M then M is -semiparallel if and only if h(y, φz) = β h(y, Z) provided α = β. α Proof. Let M be an invariant submanifold of a (ST S) 3 manifold M then M is - semiparallel then from (4.11) we get R (X, Y )( h)(z, U, W ) ( h)(r(x, Y )Z, U, W ) ( h)(z, R(X, Y )U, W ) ( h)(z, U, R(X, Y )W ) = 0. Putting X = U = ξ in (7.), we obtain (7.1) R (ξ, Y )( h)(z, ξ, W ) ( h)(r(ξ, Y )Z, ξ, W) ( h)(z, R(ξ, Y )ξ, W) ( h)(z, ξ, R(ξ, Y )W ) = 0. (7.) By virtue of (.9), (.17), (.11) and (4.7), we have the following ( h)(z, ξ, W ) = ( Z h)(ξ, W) = Z (h(ξ, W)) h( Zξ,W) h(ξ, Z W) = εαh(φz, W) βδh(z, W) (7.3) ( h)(r(ξ, Y )Z, ξ, W ) = ( R(ξ,Y)Z h)(ξ, W) = R(ξ,Y)Z (h(ξ, W)) h( R(ξ,Y)Zξ,W) h(ξ, R(ξ,Y)Z W) and = (3 ε)(α β )η(z)[εαh(φy, W) βδh(y, W)] ( h)(z, R(ξ, Y )ξ, W ) = ( Z h)(r(ξ, Y )ξ, W) = Z (h(r(ξ, Y )ξ, W)) h( ZR(ξ,Y)ξ,W) h(r(ξ, Y )ξ, Z W) = (3 ε)(α β )[ Z h(y, W) + h( Z Y, W) η(y )h( Z ξ,w)+ h(y, Z W) ( h)(z, ξ, R(ξ, Y )W ) = ( Z h)(ξ, R(ξ, Y )W) = Z (h(ξ, R(ξ, Y )W)) h( Zξ,R(ξ,Y)W) h(ξ, Z R(ξ, Y)W) (7.4) (7.5) = (3 ε)(α β )η(w)[εαh(φz, Y ) βδh(z, Y )]. (7.6)

9 Invariant submanifolds of special trans-sasakian structure 1191 Using (7.3),(7.4),(7.5),(7.6) in (7.), we get R (ξ, Y )[εαh(φz, W ) βδh(z, W)]+[(3 ε)(α β )] [ η(z)(εαh(φy, W ) βδh(y, W)) + Z h(y, W) h( Z Y, W) +η(y )h( Z ξ,w) h(y, Z W)+ η(w)[εαh(φz, Y ) βδh(z, Y )]] = 0. (7.7) Putting W = ξ in (7.7), we obtain h(y, φz) = βδ h(y, Z), (7.8) εα if (α β ) = 0. Hence the proof. 8. Invariant submanifolds of (ST S) 3 -manifolds satisfying P(X,Y) h = 0 and P(X,Y) h = 0 Theorem 8.1. Let M be an invariant submanifold of a (ST S) 3 -manifold M such that r = 4A, then P(X,Y) h = 0 holds on M if and only if M is totally geodesic. Proof. Let M be an invariant submanifold of (ST S) 3 -manifold M satisfying P(X,Y) h = 0. We have from (4.1) that R (X, Y )h(z, U) h(p (X, Y )Z, U) h(z, P (X, Y )U) = 0. (8.1) Setting X = U = ξ in (8.1) and using (3.8)and (.11), we obtain h(z, P (ξ, Y )ξ)) = 0. (8.) By virtue of (3.9) it follows from (8.) that [ εr ] Aε h(z, Y ) = 0 (8.3) It implies that Hence the proof. h(z, Y ) = 0, provided r = 4A. (8.4) Theorem 8.. Let M be an invariant submanifold of a (ST S) 3 -manifold M such that r = 4A and β = 0, then P(X,Y) h = 0 holds on M if and only if M is totally geodesic. Proof. Let M be an invariant submanifold of (ST S) 3 -manifold M satisfying P(X,Y) h = 0. We have from (4.13) that R (X, Y )( h)(z, U, W) ( h)(p (X, Y )Z, U, W ) ( h)(z, P (X, Y )U, W ) ( h)(z, U, P (X, Y )W ) = 0. (8.5)

10 119 Shivaprasanna G.S., Bhavya K. and Somashekhara G. Putting X = U = ξ in (8.5), we obtain R (ξ, Y )( h)(z, ξ, W) ( h)(p (ξ, Y )Z, ξ, W) ( h)(z, P (ξ, Y )ξ, W) ( h)(z, ξ, P (ξ, Y )W) = 0. (8.6) By virtue of (.11),(4.7),(3.8) and (3.9), we get ( h)(p (ξ, Y )Z, ξ, W ) = ( P(ξ,Y)Z h)(ξ, W) = P(ξ,Y)Z (h(ξ, W)) h( P(ξ,Y)Zξ,W) h(ξ, P(ξ,Y)Z W) = h( P(ξ,Y)Z ξ,w) = β (r 4A)h(Y, W)η(Z). (8.7) and ( h)(z, P (ξ, Y )ξ, W ) = ( Z h)(p (ξ, Y )ξ, W) = Z (h(p (ξ, Y )ξ, W) h( ZP(ξ,Y)ξ,W) h(p (ξ, Y )ξ, Z W) = B[(ε 1)η(Y ) ( h)(z, ξ, P (ξ, Y )W ) = ( Z h)(ξ, P (ξ, Y )W) (3 ε) ]h( Z ξ,w). = Z (h(ξ, P (ξ, Y )W)) h( Zξ,P(ξ,Y)W) h(ξ, Z P(ξ,Y)W) = h( Z ξ,p(ξ,y)w) = 0. (8.8) (8.9) In view of (8.7),(8.8)and(8.9), we have from (8.6) that R (ξ, Y )[εαh(φz, W ) βδh(z, W)]+ β (r 4A)η(Z)h(Y, W) (3 ε) + B[(ε 1)η(Y ) ]h( Z ξ,w) = 0. Putting Z = ξ in (8.10), we obtain (8.10) h(y, W ) = 0, provided β = 0and r = 4A. (8.11) Hence the proof. References [1] Bejancu A. and Duggal K.L., Real hypersurfaces of idefinite Kahler manifolds, Int. J. Math. Math. Sci., 16, No. 3(1993),

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