Semi-Slant Warped Product Submanifolds of a Kenmotsu Manifold

We study semi-slant warped product submanifolds of a Kenmotsu manifold. We obtain a characterization for warped product submanifolds in terms of warping function and shape operator and finally proved an inequality for squared norm of second fundamental form.


Introduction
In 1 Tanno classified the connected almost contact metric manifold whose automorphism group has maximum dimension, there are three classes: a homogeneous normal contact Riemannian manifolds with constant φ holomorphic sectional curvature if the sectional curvature of the plane section contains ξ, say C X, ξ > 0; b global Riemannian product of a line or a circle and Kaehlerian manifold with constant holomorphic sectional curvature, C X, ξ 0; c a warped product space R × f C n , if C X, ξ < 0.
Manifolds of class a are characterized by some tensorial equations, it has a Sasakian structure and manifolds of class b are characterized by some tensor equations called Cosymplectic manifolds.Kenmotsu 2 obtained some tensorial equations to Characterize manifolds of class c , these manifolds are called Kenmotsu manifolds.
The notion of semi-slant submanifolds of almost Hermitian manifolds was introduced by Papaghiuc 3 after that cabrerizo et al. 4 defined and studied semi-slant submanifolds in the setting of almost contact manifolds.

Preliminaries
A 2n 1 dimensional C ∞ manifold M is said to have an almost contact structure if there exist on M a tensor field φ of type 1, 1 , a vector field ξ, and 1-form η satisfying the following properties:

2.1
There always exists a Riemannian metric g on an almost contact manifold M satisfying the following conditions: where X, Y are vector fields on M.
An almost contact metric structure φ, ξ, η, g is said to be Kenmotsu manifold, if it satisfies the following tensorial equation 2 : for any X, Y ∈ T M, where T M is the tangent bundle of M and ∇ denotes the Riemannian connection of the metric g.Moreover, for a Kenmotsu manifold Let M be a submanifold of an almost contact metric manifold M with induced metric g and if ∇ and ∇ ⊥ are the induced connection on the tangent bundle TM and the normal bundle T ⊥ M of M, respectively, then Gauss and Weingarten formulae are given by for each X, Y ∈ TM and N ∈ T ⊥ M, where h and A N are the second fundamental form and the shape operator, respectively, for the immersion of M into M and they are related as where g denotes the Riemannian metric on M as well as on M.
For any X ∈ TM, we write where PX is the tangential component and FX is the normal component of φX.
Similarly, for any N ∈ T ⊥ M, we write where tN is the tangential component and fN is the normal component of φN.The covariant derivatives of the tensor field P and F are defined as 2.9 From 2.3 , 2.5 , 2.7 and 2.8 we have where λ −cos 2 θ.Thus, one has the following consequences of above formulae: A submanifold M of M is said to be semi-slant submanifold of an almost contact manifold M if there exist two orthogonal complementary distributions D T and It is straight forward to see that semi-invariant submanifolds and slant submanifolds are semi-slant submanifolds with θ π/2 and D T {0}, respectively.
If μ is invariant subspace under φ of the normal bundle T ⊥ M, then in the case of semislant submanifold, the normal bundle T ⊥ M can be decomposed as 2.15 A semi-slant submanifold M is called a semi-slant product if the distributions D T and D θ are 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 semi-slant product submanifold, one can consider warped product of manifolds which are defined as Definition 2.3.Let B, g B and F, g F be two Riemannian manifolds with Riemannian metric g B and g F , respectively, and f be a positive differentiable function on B. The warped product of B and F is the Riemannian manifold B × F, g , where

2.16
For a warped product manifold N 1 × f 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 T and D θ , respectively.

The warped product manifold
2.17 Bishop and O'Neill 5 proved the following.
∇f is the gradient of f and is defined as for all X ∈ TM.
Corollary 2.5.On a warped product manifold M N 1 × f N 2 , the following statements hold: Throughout, one denotes by N T and N θ an invariant and a slant submanifold, respectively, of an almost contact metric manifold M.
Khan et al. 14 proved the following corollary.
Corollary 2.6.Let M be a Kenmotsu manifold and N 1 and N 2 be any Riemannian submanifolds of M, then there do not exist a warped product submanifold Thus, one assumes that the structure vector field ξ is tangential to N 1 of a warped product submanifold In this paper we will consider the warped product of the type N θ × f N T and N T × f N θ .The warped product of the type N θ × f N T is called warped product semi-slant submanifolds; this type of warped product is studied by Atc ¸eken 10 , they proved that the warped product N θ × f N T does not exist.The warped product of the type N T × f N θ is called semi-slant warped product; these submanifolds were studied by Siraj-Uddin et al. 11 and they proved the following Lemma Lemma 2.7.Let M N T × f N θ be warped product semi-slant submanifold of a Kenmotsu manifold M such that ξ is tangent to N T , where N T and N θ are invariant and proper slant submanifolds of M. then for any X ∈ TN T and Z ∈ TN θ .

Mathematical Problems in Engineering
Replacing X by PX in part ii of above lemma one has

Semi-Slant Warped Product Submanifolds
Throughout this section we will study the warped product of the type N T × f N θ , for these submanifolds by Theorem 2.4 we have Proof.From 2.10 and 3.1 we have A FZ X th X, Z 0.

3.5
Similarly, PX ln fZ − X ln fP Z th X, Z − η X PZ, 3.6 from 3.5 and 3.6 , we get taking inner product with W ∈ TN θ , we have From Lemma 3.1 and 3.8 we get the desired result.
Conversely, let M be a semi-slant submanifold of M satisfying the hypothesis of the theorem, then for any X, Y ∈ D T ⊕ ξ and Z ∈ D θ g h X, Y , FZ 0, 3.9 Further, suppose N θ be a leaf of D θ and h θ be second fundamental form of the immersion of N θ in M, then for any X ∈ D T ⊕ ξ and Z ∈ D θ , we have g h θ Z, Z , φX g ∇ Z Z, φX , 3.11 using 2.7 and 2.5 , the above equation yields g h θ Z, Z , φX g ∇ Z PZ, X g A FZ Z, X , 3.12 applying 3.4 , we get Replacing X by PX, the above equation gives From above equation it is easy to derive that is, N θ is totally umbilical and as Zα 0, for all Z ∈ D θ , ∇μ is defined on N T , this mean that mean curvature vector of N θ is parallel, that is, the leaves of D θ are extrinsic spheres in Mathematical Problems in Engineering M. Hence by virtue of result of 15 which says that if the tangent bundle of a Riemannian manifold M splits into an orthogonal sum TM E 0 ⊕ E 1 of nontrivial vector subbundles such that E 1 is spherical and its orthogonal complement E 0 is autoparallel, then the manifold M is locally isometric to a warped product M 0 × f M 1 , we can say M is locally semi-slant warped product submanifold N T × f N θ , where the warping function f e α .Let us denote by D T and D θ the tangent bundles on N T and N θ , respectively, and let {X 0 ξ, X 1 , . . ., X p , X p 1 φX 1 , . . ., X 2p φX p } and {Z 1 , . . ., Z q , Z q 1 PZ 1 , . . ., Z 2q PZ q } be local orthonormal frames of vector fields on N T and N θ , respectively, with 2p and 2q being real dimension.Since h X, ξ 0 for all X ∈ TM, then the second fundamental form can be written as

3.16
Now, on a semi-slant warped product submanifold of a Kenmotsu manifold, we prove the following.

Theorem 3.3. Let M
N T × f N θ be a semi-slant warped product submanifold of a Kenmotsu manifold M with N T and N θ invariant and slant submanifolds, respectively, of M. If η X ≥ 2X ln f for all X ∈ TN T , then the squared norm of the second fundamental form h satisfies where ∇ ln f is the gradient of ln f and 2q is the dimension N θ .
Proof.In view of the decomposition 2.15 , we may write

3.24
Since we have choose the orthonormal frame of vector fields on D θ as {Z 1 , . . ., Z q , Z q 1 PZ 1 , . . ., Z 2q PZ q }, then the second term in the right-hand side of 3.24 is written as

3.25
From part i of Lemma 2.7, the first two terms of above equation can be written as 2q X i ln f − η X i 2 cos 4 θ .

3.26
In account of to hypothesis η X i ≥ 2X i ln f the above expression is greater than equal to the following term: 4qcsc 2 θ ∇ ln f 2 cos 4 θ.

3.28
The inequality 3.17 follows from 3.16 and 3.28 .The equality holds if h D T , D T 0, h D θ , D θ 0, h PX, Z is orthogonal to FZ and FP Z for all X ∈ D T and Z ∈ D θ and η X 2X ln f.

Conclusion
In this paper we study nontrivial warped product submanifolds of a Kenmotsu manifold and in this study there emerge natural problems of finding the estimates of the squared norm of second fundamental form and to find the relation between shape operator and warping function.This study predict the geometric behavior of underlying warped product submanifolds.Further, as it is known that the warping function of a warped product manifold is a solution of some partial differential equations c.f., 8 and most of physical phenomenon is described by partial differential equations.We hope that our study may find applications in physics as well as in engineering.
Let M be a submanifold of an almost contact manifold M, such that ξ ∈ TM then M is slant submanifold if and only if there exists a constant λ ∈ 0, 1 such that ln fZ, 3.1 for any X ∈ TN T and Z ∈ TN θ .
for any X, Y ∈ TN T and Z ∈ TN θ .Proof.As N T is totally geodesic in M then ∇ X P Y ∈ TN T and therefore by formula 2.10 :∇ X P Y th X, Y − g X, P Y ξ − η YPX, 3.3 taking inner product with Z ∈ TN θ we get 3.2 .Now we have the following Characterization.Theorem 3.2.A semi-slant submanifold M of a Kenmotsu manifold M with integrable invariant distribution D T ⊕ ξ and integrable slant distribution D θ is locally a semi-slant warped product if and only if ∇ Z PZ ∈ D θ and there exists a C ∞ -function α on M with Zα 0, TM.In view of above formulae we haveg h FD θ PX i , Z r , h FD θ PX i , Z r g h r PX i , Z r FZ r , h r PX i , Z r FZ r g h s PX i , Z r FZ r , h s PX i , Z r FZ r .FD θ PX i , Z r , h FD θ PX i , Z r h r PX i , Z r X i ln f sin 2 θ