Another class of warped product skew CR-submanifolds of Kenmotsu manifolds

Recently, Naghi et al. \cite{NAGHI} studied warped product skew CR-submanifold of the form $M_1\times_fM_\bot$ of order $1$ of a Kenmotsu manifold $\bar{M}$ such that $M_1=M_T\times M_\theta$, where $M_T$, $M_\bot$ and $M_\theta$ are invariant, anti-invariant and proper slant submanifolds of $\bar{M}$. The present paper deals with the study of warped product submanifolds by interchanging the two factors $M_T$ and $M_\bot$, i.e, the warped products of the form $M_2\times_fM_T$ such that $M_2=M_\bot\times M_\theta$. The existence of such warped product is ensured by an example and then we characterize such warped product submanifold. A lower bounds of the square norm of second fundamental form is derived with sharp relation, whose equality case is also considered.


Introduction
In 1986, Bejancu [4] introduced the notion of CR-Submanifolds. This family of submanifolds was generalized by Chen [9] as slant submanifolds. Then a more generalization is given as semi-slant submanifolds by Papaghiuc [33]. Next, Cabrerizo et al. [7] defined and studied bi-slant submamifolds and simultanously gave the notion of pseudo-slant submanifolds. The contact version of slant, semi slant and pseudo-slant submanifolds are studied in [28], [7] and [24], respectively. As a generalization of all these class of submanifolds, Ronsse [34] introduced the notion of skew CR-submanifolds of Kaehler manifolds.
Warped product skew CR-submanifolds of Kaehler manifold was studied by Sahin [35] and in [13]  in section 2, some preliminaries are given, section 3 is dedicated to the study of skew CR-submanifold of Kenmotsu manifold, in section 4, we provide an example of warped product skew CR-submanifolds of the form M 2 × f M T and some basic results of such type of submanifolds are obtained, a characterization of skew CR-warped product of the form M 2 × f M T is obtained in section 5. In section 6, we have established two inequalities on a warped product skew CR-submanifold M = M 2 × f M T of a Kenmotsu manifoldM.

Preliminaries
In [36] Tanno classified connected almost contact metric manifolds whose automorphism groups possess the maximum dimension. For such a manifold, the sectional curvature of plane sections containing ξ is a constant, say c. He proved that they could be divided into three classes: (i) homogeneous normal contact Riemannian manifolds with c > 0, (ii) global Riemannian products of a line or a circle with a Kähler manifold of constant holomorphic sectional curvature if c = 0 and (iii) a warped product space R × f C n if c < 0.
Kenmotsu [23] characterized the differential geometric properties of the manifolds of class (iii) which are nowadays called Kenmotsu manifolds and later studied by several authors ([16]- [18]) etc.
An odd dimensional smooth manifoldM 2m+1 is said to be an almost contact metric manifold [5] if it admits a (1, 1) tensor field φ, a vector field ξ, an 1-form η and a Riemannian metric which satisfy for all vector fields X, Y onM. An almost contact metric manifoldM 2m+1 (φ, ξ, η, ) is said to be Kenmotsu manifold if the following conditions hold [23]: where∇ denotes the Riemannian connection of . Let M be an n-dimensional submanifold of a Kenmotsu manifoldM. Throughout the paper we assume that the submanifold M ofM is tangent to the structure vector field ξ. Let ∇ and ∇ ⊥ be the induced connections on the tangent bundle TM and the normal bundle T ⊥ M of M respectively. Then the Gauss and Weingarten formulae are given bȳ and∇ for all X, Y ∈ Γ(TM) and N ∈ Γ(T ⊥ M), where h and A N are second fundamental form and the shape operator (corresponding to the normal vector field N) respectively for the immersion of M intoM and they are related by (h(X, Y), N) = (A N X, Y) for any X, Y ∈ Γ(TM) and N ∈ Γ(T ⊥ M), where g is the Riemannian metric onM as well as on M.
The mean curvature H of M is given by H = 1 n trace h. A submanifold M of a Kenmotsu manifoldM is said to be totally umbilical if h(X, Y) = (X, Y)H for any X, Y ∈ Γ(TM). If h(X, Y) = 0 for all X, Y ∈ Γ(TM), then M is totally geodesic and if H = 0 then M is minimal inM.
Let {e 1 , · · · , e n } be an orthonormal basis of the tangent bundle TM and {e n+1 , · · · , e 2m+1 } be that of the normal bundle T ⊥ M. Set h r ij = (h(e i , e j ), e r ) and h 2 = (h(e i , e j ), h(e i , e j )), for i, j ∈ {1, · · · , n} and r ∈ {n + 1, · · · , 2m + 1}. For a differentiable function f on M, the gradient ∇ f is defined by for any X ∈ Γ(TM). As a consequence, we get For any X ∈ Γ(TM) and N ∈ Γ(T ⊥ M), we can write where PX, bN are the tangential components and QX, cN are the normal components.
A submanifold M of an almost contact metric manifoldM is said to be invariant if φ(T p M) ⊆ T p M and anti-invariant if φ(T p M) ⊆ T ⊥ p M for every p ∈ M. A submanifold M of an almost contact metric manifoldM is said to be slant if for each non-zero vector X ∈ T p M, the angle θ between φX and T p M is a constant, i.e. it does not depend on the choice of p ∈ M. Invariant and anti-invariant submanifolds are particular cases of slant submanifolds with slant angles θ = 0 and π 2 respectively.
If M is a slant submanifold of an almost contact metric manifoldM, the following relation holds [38]: Definition 2.2. [6] Let (N 1 , 1 ) and (N 2 , 2 ) be two Riemannian manifolds with Riemannian metric 1 and 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 A warped product manifold N 1 × f N 2 is said to be trivial if the warping function f is constant. For a warped product manifold M = N 1 × f N 2 , we have [6] ∇ U X = ∇ X U = (X ln f )U (15) for any X, Y ∈ Γ(TN 1 ) and U ∈ Γ(TN 2 ). We now recall the following: Theorem 2.3. (Hiepko's Theorem, see [15]). Let D 1 and D 2 be two orthogonal distribution on a Riemannian manifold M. Suppose that D 1 and D 2 both are involutive such that D 1 is a totally geodesic foliation and D 2 is a spherical foliation. Then M is locally isometric to a non-trivial warped product M 1 × f M 2 , where M 1 and M 2 are integral manifolds of D 1 and D 2 , respectively.

Skew CR-submanifolds of Kenmotsu manifolds
Let M be a submanifold of a Kenmotsu manifoldM. First from [34], we recall the definition of skew CRsubmanifolds. Throughout the paper we consider the structure vector field ξ is tangent to the submanifold otherwise the submanifold is C-totally real [14].
For any X and Y in T p M, we have (PX, Y) = − (X, PY). Hence it follows that P 2 is symmetric operator on the tangent space TM, for all p ∈ M. Therefore the eigen values are real and it is diagonalizable. Moreover its eigen values are bounded by −1 and 0. For each p ∈ M, we may set where I is the identity transformation and λ(p) ∈ [0, 1] such that λ 2 (p) is an eigen value of P 2 p . We note that D 1 p = kerQ and D 0 p = kerP. D 1 p is the maximal φ-invariant subspace of T p M and D 0 p is the maximal φ-anti-invariant subspace of T p M. From now on, we denote the distributions D 1 and D 0 by D T ⊕ < ξ > and D ⊥ , respectively. Since P 2 p is symmetric and diagonalizable, if −λ 2 1 (p), · · · , −λ 2 k (p) are the eigenvalues of P 2 at p ∈ M, then T p M can be decomposed as direct sum of mutually orthogonal eigen spaces, i.e. A submanifold M ofM is said to be proper skew CR-submanifold of order 1 if M is a skew CRsubmanifold with k = 1 and λ 1 is constant. In that case, the tangent bundle of M is decomposed as The normal bundle T ⊥ M of a skew CR-submanifold M is decomposed as Now for the sake of further study we give the following useful results.

Lemma 3.3.
Let M be a proper skew CR-submanifold of order 1 of a Kenmotsu manifoldM such that ξ ∈ Γ(D ⊥ ⊕D θ ), then we have Proof. For every X ∈ Γ(D T ) and Z, W ∈ Γ(D ⊥ ), we have Using (5), (7) and orthogonality of vector fields in the above equation, we get (20). Also, for X ∈ Γ(D T ), Z ∈ Γ(D ⊥ ) and U ∈ Γ(D θ ), we have Using (7), (12) and the symmetry of shape operator in the above equation, we obtain By virtue of (5) and (7), the above equation yields from which the relation (22) follows. Again we have Using (7), (12) and the symmetry of shape operator in the above equation, we get from which we get (23). For every X ∈ Γ(D T ), Z ∈ Γ(D ⊥ ) and U ∈ Γ(D θ ), we have In view of (7), (13) and the symmetry of shape operator, the above equation reduces to from which the relation (24) follows.

Warped product skew CR
Now, we consider a submanifold M ofM defined by the immersion χ as follows: Then the local orthonormal frame of TM is spanned by the following: Also, we have and (h(X, U), QV) = 0 for every X ∈ Γ(M T ), Z, W ∈ Γ(M ⊥ ) and U, V ∈ Γ(M θ ).
Proof. For every X, Y ∈ Γ(M T ) and Z ∈ Γ(M ⊥ ), we have Using (15) in the above equation, we obtain from which the relation (32) follows. Also, for every X, Y ∈ Γ(M T ) and U ∈ Γ(M θ ), we have Using (5) and (15) in the above equation, we obtain from which the relation (33) follows. Also, replacing U by PU in (33) and using (12), we get (34). Now, replacing X by φX and Y by φY in (32), we obtain the following: Also, replacing X by φX and Y by φY in (33), we get the following: and Proof. The relation (i) follows from (35) and (36). The relation (ii) follows from (32) and (37). The relation (iii) follows from (38) and (39). The relation (iv) follows from (33) and (40).   (27), we have
Finally, we show that H T is parallel with respect to the normal connection D N of M T in M. We take E ∈ Γ(D ⊥ ⊕ D θ ⊕ {ξ}) and X ∈ Γ(D T ), then we have where ∇ ⊥ µ, ∇ θ µ and ∇ ξ µ are the gradient components of µ on M along D ⊥ , D θ and {ξ} respectively. Then by the property of Riemannian metric, the above equation reduces to since (Xµ) = 0, for any X ∈ Γ(D T ) and ∇ Z ∇ ⊥ µ + ∇ U ∇ θ µ + ∇ ξ ∇ ξ µ = ∇ E ∇µ is orthogonal to D T for any E ∈ Γ(D ⊥ ⊕ D θ ⊕ {ξ}) as ∇µ is the gradient along M 2 and M 2 is totally geodesic inM. Therefore, the mean curvature vector H T of M T is parallel. Thus, M T is an extrinsic sphere in M. Hence by Theorem 2.3, M is locally a warped product submanifold. Thus the proof is complete.

Generalized inequalities on warped product skew CR-submanifolds
In this section, we establish two inequalities on a warped product skew CR-submanifold M = M 2 × f M T of a Kenmotsu manifoldM such that M 2 = M ⊥ × M θ . We take dimM T = 2p, dimM ⊥ = q, dimM θ = 2s + 1 and their corresponding tangent spaces are D T , D ⊥ and D θ ⊕ {ξ} respectively.
Decomposing the above relation for our constructed frames, we get (h(e i , e j ), e r ) 2 .
Now, by Proposition 4.3, the tenth, eleventh, thirteenth and fourteenth terms of (64) are equal to zero. Also, we can not find any relation for a warped product in the form (h(E, F), ν) for any E, F ∈ Γ(TM). So, leaving the positive third, sixth, ninth, twelfth, fifteenth and eighteenth terms of (64) we get (h(e i , e j ),ẽ r ) 2 .
Also, leaving fifth and sixth term of (65), we get h(D θ , D θ ) ⊥ φD ⊥ and h(D θ , D θ ) ⊥ QD θ respectively. Therefore, From (77) and (78), we obtain Next by leaving the twelfth term of (64), we get h(D ⊥ , D T ) ⊥ ν. If the equality of (85) holds, then M 2 is totally geodesic and M T is totally umbilical inM.

Remark:
If we take dim M θ = 0 in a warped product skew CR-submanifold M = M 2 × f M T of a Kenmotsu manifoldM such that M 2 = M ⊥ × M θ , then it turns into CR-warped product M = M ⊥ × f M T which was studied in [40]. Therefore, Theorem 5.1 and Theorem 6.2 are the generalizations of results of [40] as follows:  [40]) LetM be a (2m + 1)-dimensional Kenmotsu manifold and M = M ⊥ × f M T an n-dimensional contact CR-warped product submanifold, such that M ⊥ is a (q + 1)-dimensional anti-invariant submanifold tangent to ξ and M T is a 2p-dimensional invariant submanifold ofM, then the squared norm of the second fundamental form of M satisfies where ∇ ⊥ ln f is the gradient of ln f . If the equality of (86) holds, then M ⊥ is totally geodesic and M T is totally umbilical inM.