Geometry of warped product semi-slant submanifolds of Kenmotsu manifolds

In this paper, we study semi-slant submanifolds and their warped products in Kenmotsu manifolds. The existence of such warped products in Kenmotsu manifolds is shown by an example and a characterization. A sharp relation is obtained as a lower bound of the squared norm of second fundamental form in terms of the warping function and the slant angle. The equality case is also considered in this paper. Finally, we provide some applications of our derived results.


Introduction
Warped product manifolds were studied by Bishop, and O'Neill in 1969 as a natural generalization of the Riemannian product manifolds [4]. Later on, the geometrical aspects of these manifolds have been studied by many researchers [7,11]. Recently, B.Y. Chen [7] has given the idea of warped product submanifolds. Motivated by Chen's papers, many geometers studied warped product submanifolds for different structures on Riemannian manifolds [1,10,12].
On the other hand, in [6] the authors studied slant immersions in K-contact and Sasakian manifolds. They introduced many interesting examples of slant submanifolds in both almost contact metric manifolds and Sasakian manifolds. They characterized slant submanifolds by means of the covariant derivative of the square of the tangent projection on the submanifold of almost contact structure of a K-contact manifold. Later on, in [5], they have defined and studied semi-slant submanifolds of Sasakian manifolds.
Recently, Atceken studied warped product semi-slant submanifolds in Kenmotsu manifolds and Kenmotsu space forms, he has shown the non-existence cases of the warped product semi-slant submanifolds in a Kenmotsu manifold [2,3]. Later on, S. Uddin et al. studied warped product semi-slant submanifolds of Kenmotsu manifolds and showed that the warped product submanifolds exist except the case when the structure vector field ξ is tangent to the fiber [12]. They have obtained some existence results which we will use in this paper. In this paper, we study semi-slant submanifolds in brief not in details as our aim is to dicuss the warped products. We prove the existence of warped product semi-slant submanifolds by a characterization and obtain an inequality for the squared norm of the second fundamental form in terms of the warping function and the slant angle for the warped product semi-slant submanifolds of a Kenmotsu manifold.

Preliminaries
Let M be an almost contact metric manifold with structure (ϕ, ξ, η, g) where ϕ is a (1, 1) tensor field, ξ a vector field, η is a 1-form and g is a Riemannian 2010 AMS Mathematics Subject Classification: 53C40, 53C42, 53B25. metric on M satisfying the following properties If in addition to the above relations, holds, then M is said to be Kenmotsu manifold, where ∇ is the Levi-Civita connection of g. We shall use the symbol Γ(TM ) to denote the Lie algebra of vector fields on the manifold M . Let M be submanifold of an almost contact metric manifold M with induced metric g and if ∇ and ∇ ⊥ are the induced connections on the tangent bundle T M and the normal bundle T ⊥ M of M , respectively then Gauss and Weingarten formulae are given by for each X, Y ∈ Γ(T M ) and N ∈ Γ(T ⊥ M ), where h and A N are the second fundamental form and the shape operator (corresponding to the normal vector field N ) respectively for the immersion of M into M . They are related as For any X ∈ Γ(T M ) and N ∈ Γ(T ⊥ M ) , we write where P X and tN are the tangential components of ϕX and ϕN respectively and F X and f N are the normal components of ϕX and ϕN , respectively. The submanifold M is said to be invariant if F is identically zero, that is, ϕX ∈ Γ(T M ) for any X ∈ Γ(T M ). On the other hand M is said to be antiinvariant if P is identically zero, that is, ϕX ∈ Γ(T ⊥ M ), for any X ∈ Γ(T M ). For a Riemannian submanifold M of a Kenmotsu manifold M , we have and are the covariant derivative of the tensor field P and F , respectively.
Let M be a submanifold tangent to the structure vector field ξ isometrically immersed into an almost contact metric manifold M . Then M is said to be contact CR-submanifold if there exists a pair of orthogonal distributions D : where ξ is the 1-dimensional distribution spanned by the structure vector field ξ.
Invariant and anti-invariant submanifolds are the special cases of a contact CRsubmanifold. If we denote the dimensions of the distributions D and D ⊥ by d 1 and d 2 , respectively. Then is M is invariant (resp. anti-invariant) if d 2 = 0 (resp. d 1 = 0). There is another class of submanifolds that is called the slant submanifold. For each non zero vector X tangent to M at x, such that X is not proportional to ξ, we denotes by θ(X), the angle between ϕX and P X.
M is said to be slant [6] if the angle θ(X) is constant for all X ∈ T M − {ξ} and x ∈ M . The angle θ is called slant angle or Wirtinger angle. Obviously if θ = 0, M is invariant and if θ = π/2, M is an anti-invariant submanifold. If the slant angle of M is different from 0 and π/2 then it is called proper slant.
A characterization of slant submanifolds is given by the following result.
Theorem 2.1 [6]. Let M be a submanifold of an almost contact metric manifold Furthermore, in such case, if θ is slant angle, then λ = cos 2 θ. Following relations are straight forward consequence of equation (2.9) for any X, Y tangent to M.

Semi-slant submanifolds
Semi-slant submanifolds were defined and studied by N. Papaghiuc [11] as a natural generalization of CR-submanifolds of almost Hermitian manifolds in terms of slant distribution. Later on, Cabrerizo et al. [4] studied these submanifolds in contact setting. They defined these submanifolds as follows: Definition 3.1 [5]. A Riemannian submanifold M of an almost contact manifold M is said to be a semi-slant submanifold if there exist two orthogonal distributions D and D θ such that T M = D ⊕ D θ ⊕ ξ , the distribution D is invariant i.e., ϕD = D and the distribution D θ is slant with slant angle θ = π 2 . If we denote the dimensions of D and D θ by d 1 and d 2 respectively, then it is clear that contact CR-submanifolds and slant submanifolds are semi-slant submanifolds with θ = π 2 and d 1 = 0, respectively. A semi-slant submanifold M is said to be mixed geodesic if h(X, Z) = 0, for any X ∈ Γ(D) and Z ∈ Γ(D θ ). Moreover, if ν is the ϕ−invariant subspace of the normal bundle T ⊥ M , then in case of semi-slant submanifold, the normal bundle T ⊥ M can be decomposed as In this section, we prove the integrability conditions of involved distributions of semi-slant submanifolds of a Kenmotsu manifold which we required for the characterization of warped products.
Proof. From the definition of Lie bracket and (2.2), we have for any X, Y ∈ Γ(D ⊕ ξ ) and Z ∈ Γ(D θ ). Using the definition of covariant derivative of ϕ and the relation (2.6)-(a), we get Then from (2.3), (2.4) and the fact that ϕY and P Z are orthogonal vector fields, we derive Again using (2.6)-(a) and then by (2.9) and (2.5), we obtain By the orthogonality of two distributions ones get Again, by the definition of Lie bracket, we derive Finally, the above equation can be written as Hence the result follows from the last relation. Now, we prove the integrability of the slant distribution. for any X ∈ Γ(D ⊕ ξ ) and Z, W ∈ Γ(D θ ).

warped product semi-slant submanifolds
The idea of warped product manifolds was given by Bishop and O'Niell [4]. They defined these manifolds as follows: Definition 4.1 [4] . Let (N 1 , g 1 ) and (N 2 , g 2 ) be two Riemannian manifolds with Riemann metric g 1 and g 2 , respectively, and f be a positive smooth function on N 1 . The warped product of N 1 and N 2 is the Riemannian manifold M = For a warped product manifold M = N 1 × f N 2 = (N 1 × N 2 , g) is said to be trivial if the warping function f is constant. More explicitly, if the vector fields X and Y are tangent to M = N 1 × f N 2 at (p, q), then g(X, Y ) = g 1 (π 1 * X, π 1 * Y ) + f 2 (p)g 2 (π 2 * X, π 2 * Y ) where π 1 and π 2 are the canonical projections of M = N 1 × N 2 onto N 1 and N 2 respectively and * is the symbol for the tangent map. Now, let us recall the following results on warped products for later use.
Lemma 4.1 [4]. Let M = N 1 × f N 2 be a warped product manifold with the warping function f. Then for any X, Y ∈ Γ(T N 1 ) and Z, W ∈ Γ(T N 2 ), where ∇ and ∇ N2 denote the Levi-Civita connections on M and N 2 , respectively and ∇f is the gradient of f defined by g(∇f, X) = X(f ), for any X ∈ Γ(T M ). As a consequence, we have the equality for the orthonormal frame {e 1 , · · · e n } of the tangent space of M .
The following result is useful to prove our main theorem. for any X ∈ Γ(T N T ) and Z ∈ Γ(T N θ ).
In the above lemma, if we replace X by ϕX in the third part, then we have Also, for any X, Y ∈ Γ(T N T ) and Z ∈ Γ(T N θ ), we have Using (2.6)-(a), we obtain Then from (2.4) and Lemma 4.1 (ii), the first term of right hand side is identically zero. Also, by (2.3) and the fact that ξ is tangent to N T , the second term vanishes identically, hence g(h(X, Y ), F Z) = 0.  Conversely, if M is a semi-slant submanifold of a Kenmotsu manifold M such that (4.5) holds, then by the Theorem 3.1, the distribution D⊕ < ξ > is integrable if and only if for any X, Y ∈ Γ(D⊕ < ξ >) and Z ∈ Γ(D θ ). Using (4.5), we arrive at Hence D⊕ < ξ > is integrable and its leaves are totally geodesic in M . Also, from Theorem 3.2, the slant distribution D θ is integrable if and only if g(∇ W Z, X) = csc 2 θ{g(A F P W X, Z) − g(A F W ϕX, Z)} − η(X)g(Z, W ) for any Z, W ∈ Γ(D θ ) and X ∈ Γ(D⊕ < ξ >). If h θ be the second fundamental form of a leaf N θ in M , then from (4.5), we derive Thus, from the definition of gradient and last relation, we get which means that N θ is totally umbilical in M with mean curvature vector H θ = −∇µ. Now we prove that H θ is parallel corresponding to the normal connection D of N θ in M . For this consider any Y ∈ Γ(D⊕ < ξ >) and since Z(λ) = 0, for all Z ∈ Γ(D θ ) and thus ∇ Y ∇λ ∈ Γ(D⊕ < ξ >). This means that the mean curvature of N θ is parallel. Thus the leaves of D θ are totally umbilical with parallel mean curvature. Hence, by a result of Hiepko [8], M is a warped product submanifold, which completes the proof. Now, we have the following result for a mixed geodesic warped product semi-slant submanifold of a Kenmotsu manifold. Now, using the above mentioned results we are able to prove our main theorem which generalizes the Theorem 3.1 in [1].
where ∇ ln f is the gradient of the warping function ln f and 2β is the real dimension of the tangent space of N θ .
(ii) If the equality sign in (i) holds, then N T is totally geodesic submanifold and N θ is totally umbilical submanifold of M . Moreover, M is minimal submanifold of M .
Hence, to satisfy (4.2), we add and subtract the same term, we get (ξ ln f ) 2 g(e ⋆ j , e ⋆ j ) 2 .
Then from Lemma 4.2 (i), we derive for any Z, W ∈ Γ(D θ ). Since N T is totally geodesic submanifold in M , using this fact with the first condition of (4.10), we get N T is totally geodesic in M . Also, since N θ is totally umbilical in M , using this fact with (4.6) and the second condition of (4.10), we get N θ is totally umbilical in M . Moreover all conditions of (4.10) with the above fact show the minimality of M in M . This proves the theorem completely.