Warped Product Submanifolds of an S-manifold

article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we prove that there does not exist warped product of the type N θ × λ N T and N T × λ N θ of an S-manifold , where N θ and N T are the slant and invariant submanifold of an S-manifold .We observe that only warped product of the type N T × λ N ⊥ exist,with N ⊥ as a anti-invariant submanifolds of an S-manifold.Some basic results are discussed for the warped product of the type N T × λ N ⊥ and finally ,we prove an inequality for the squared norm of second fundamental form and equality case is also discussed.


Introduction
The study of semi-invariant submanifold or contact CR-submanifolds of almost contact metric manifolds was initiated by A.Bejancu [3] .In particular ,semi-invariant submanifolds of different classes of almost contact metric manifolds have also been studied (c.f., [4] , [18]).Later N.Papaghuic [18] generalized the notion further and introduced semi-slant submanifolds of almost Hermition manifolds which includes the class of slant , holomorphic ,totally real and CR-submanifolds.J.L.Cabrerizo et al. [15] have studied semi-slant submanifolds of Sasakian manifolds.For manifolds with an f -structure ,D.E Blair [9] introduced S-manifolds as the analogue of Kaehler structure in almost complex case and of Sasakian structure in the almost contact case .[20] introduced the notion of warped product manifolds.These manifolds are generaliztion of Riemannian product manifolds and occur naturally e.g.,surface of revolution is a warped product manifold ( [16] , [21]).Due to wide applications of warped product submanifolds this becomes a fascinating and interesting topic for research and many articles are available in literature .CR-warped product was introduced by B.Y.chen [7,8], he studied warped products CR-submanifolds in the setting of Kaehler manifolds and showed that there does not exist warped product CR-submanifolds of the form M ⊥ × f M T , therefore he considered warped product CR-submanifolds of the types M T × f M ⊥ and established a relation ship between the warping fuctionf and the squared norm of the second fundamental form of the CR-warped product submanifolds in Kaehler manifolds .In the available literature,many geometers have studied warped products in the setting of almost contact metric manifolds (c.f., [13] , [21] , [22]).

Preliminaries
Let (M, g) be a Riemannian manifold with dim (M ) = 2n + s.M is said to be an S-manifolds if there exists on M an f -structure f of rank 2n and s global vector fields ξ 1 , ξ 2 , ...., ξ s (structure vector fields) such that (c.f., [9]).
Moreover, for the Riemannian conection ∇ of g on an S-manifold M , the following formulas were also proved in [9] .
Throughout ,we denote by M an S-manifold ,M a submanifold of M with structure vector fields ξ 1 , ξ 2 , ..., ξ s tangent to M. h and A denote the second fundamental form and the shape opreator of the immersion of M into M respectively .If ∇ is the induced connection on M, the Gauss and Weingarten formulae of M into M are then given as follows for all vector fields X, Y on M and normal vector fields V on M ,∇ ⊥ denotes the connection on the normal bundle T M ⊥ of M. h and A are related by where the induced Riemannian metric on M is denoted by the same symbol g.Now , for any where T X and NX are the tangential and normal parts of f X respectively and tV and nV are the tangential and normal parts of f V .
The covariant derivatives ∇T and ∇N are defined by Now, Let M be an n-dimensional submanifold immersed in M.M is said to be invariant submanifold if ξ α ∈ T M for any α and f X ∈ T M, for any For a submanifold M of an S-manifold M by equation (2.4),(2.5)and (2.8), we get for each X ∈ T M. Now using equations (2.3),(2.5),(2.8),(2.10),(2.11)we get For any x ∈ M and X ∈ T X M if the vectors X and ξ α (α = 1, ..., s) are linearly independent , the angle θ(X) ∈ 0, π 2 between f X and T X M is well defined .If θ(X) does not depend on the choice of x ∈ M and X ∈ T X M ,we say that M is slant in M .The constant angle θ is called the slant angle of M in M .If θ = π 2 then M is an anti-invariant submanifold and if θ = 0 M is invariant submanifold .
Carriazo et al. [1] give the following characterization for slant submanifold of S-manifolds.Theorem 2.1 [1] Let M be a submanifold of an S-manifold M such that ξ α ∈ T M.Then, M is slant if and only if there exists a constant λ ∈ [0, 1] such that Furthermore, in such case ,if θ is the slant angle of M, then it verifies that λ = cos 2 θ.Now, we have the following Corollary,which can be proved directly by (2.16).

Corollary 2.1 Let M be a slant submanifold of an S-manifold M such that
for any X, Y ∈ T M.
A semi-slant submanifold M of an almost contact metric manifold M is a submanifold which contains two orthogonal complementary distribution D and D θ ,such that D is invariant under f and D θ is slant with slant angle θ = 0, i.e ,fD = Dand f Z makes a constant angle θ with T M for each Z ∈ D θ .In particular if θ = π 2 ,then semi-slant submanifolds reduced to CR-submanifold defined in [3] .For a semi-slant submanifold M of an S-manifold , we have The orthogonal complement of ND θ in the normal bundle T M ⊥ , is an invariant subbundle of T M ⊥ and is denoted by μ .Thus, we have (2.17) A semi-slant submanifold M is called a semi-slant product if the distributions D and D θ are involutive and parallel on M. In this case M is foliated by the leaves of these distributions.
As a generalization of the product manifolds and in particular of a semislant product submanifold , one can consider warped product of manifolds which are defined as Definition 2.1 Let (B, g B ) and (F, g F ) be two Riemannian manifolds with Riemannian metric g B and g F respectively and λ positive differentiable function on B .The warped product of B and F is the Riemannian manifold (B × F, g),where g = g B + λ 2 g F For a warped product manifold N 1 × λ N 2 ,we denote by D 1 and D 2 the distributions defined by the vectors tangent to the leaves and fibers respectively .
In other words, D 1 is obtained by the tangent vectors of N 1 via the horizontal lift and D 2 is obtained by the tangent vectors of N 2 via vertical lift .In case of semi-slant warped product submanifolds D 1 and D 2 are replaced by D and D θ respectively .
The warped product manifold [20] proved the following

R.L.Bishop and B.O'Neill
∇λ is the gradient of λ and is defined as g(∇λ, X ) = X λ, (2.18) for all X ∈ T M.

Warped Product Submanifolds of S-manifolds
Let M be an S-manifold.Throughout this section,we denote by N T an invariant submanifold of M and N θ a slant submanifold of M,with slant angle θ. [17] proved the following theorem .

Matsumoto and Mihai
a warped product submanifold of a Sasakian manifold M, where N 1 and N 2 are any submanifold of a Sasakian manifold M with ξ tangential to N 2 .Then M is a Riemannian product .
On the line of Theorem 3.1 , we can prove the following theorem Theorem 3.2 If N 1 × λ N 2 is a warped product submanifold of an S-manifold M,where N 1 and N 2 are any submanifold of an S-manifold M with ξ α tangential to N 2 .Then M is a Riemannian product .
Hence we can consider only nontrivial semi-slant warped product submanifolds as N θ × λ N T and N T × λ N θ with ξ 1 , ξ 2 , ...., ξ s tangential to N θ and N T respectively .If θ = π 2 these warped products are known as warped product contact CR-submanifolds and contact CR-warped product submanifolds respectively .
Theorem 3.3.Let M be a (2m + s)-dimensional S-manifold .Then there do not exists warped product submanifolds N θ × λ N T on M such that N θ is slant submanifold tangent to ξ 1 , ξ 2 , ..., ξ s and N T is an invariant submanifold of M.
Proof.For warped product submanifold N θ × λ N T of M with ξ α tangential to N θ .Then by Theorem 2.2 for any X ∈ T N T and Z ∈ T N θ .In particular for Z = ξ α , from equations (2.12) and (3.1) , this mean ξ α ln λ = 0 i.e.,λ is constant , hence warped product does not exists.
On the same line of Theorem 3.3 of [22] ,we can prove the following theorem.Theorem 3.4 There does not exists a warped product semi-slant submanifold of the type N T × λ N θ in an S-manifold other than a contact CR-warped submanifold.
The above theorem motivates us to study the warped product of the type N T × λ N ⊥ in S-manifolds .
Let M = N T × λ N ⊥ be a contact CR-warped product submanifold of an S-manifold M. In view of decomposition (2.17) , we may write for each X, Y ∈ T M,where h fD ⊥ (X, Y ) ∈ f D ⊥ and h μ (X, Y ) ∈ μ.If {e 1 , e 2 , ..., e n } be a local orthonormal frame of vector fields on M then we define g(h(e i , e j ), h(e i , e j )).Now we have the following proposition for the warped product of the type for any X ∈ T N T and Z, W ∈ T N ⊥ .
Proof.By Gauss formula Using equations (2.3),(2.5)and(3.1), we have Comparing tangential parts on above equation , we get Zη α (X), taking inner product with W ∈ D ⊥ on both side , we find Which proves the part (i) of the proposition .Now on comparing the normal parts Taking inner product with f W,the above equation yields Again taking inner product with f h(X, Z) in (3.3),we find which is the part (iii) of the proposition .For contact CR-warped product submanifold M = N T × λ N ⊥ of an Smanifold M, we have the following theorem.Theorem 3.5 Let M = N T × λ N ⊥ be a contact CR-warped product submanifold of an S-manifold M then (i) The squared norm of the second fundamental form satisfies where ∇ ln λ is the gradient of ln λ and q is the dimension of anti-invariant distribution .