Geometry of pointwise CR-slant warped products in Kaehler manifolds

We call a submanifold M of a Kaehler manifold M̃ a pointwise CR-slant warped product if it is a warped product, B×f Nθ, of a CR-product B = NT ×N⊥ and a proper pointwise slant submanifold Nθ with slant function θ, where NT and N⊥ are complex and totally real submanifolds of M̃ . We prove that if a pointwise CR-slant warped product B×fNθ with B = NT×N⊥ in a Kaehler manifold is weakly Dθ-totally geodesic, then it satisfies ‖σ‖2 ≥ 4s { (csc2 θ + cot2 θ)‖∇ (ln f)‖2 + (cot2 θ)‖∇⊥(ln f)‖2 } , where NT , N⊥, and Nθ are complex, totally real and proper pointwise slant submanifolds of M̃ , respectively, and s = 2 dimNθ. In this paper we also investigate the equality case of the inequality. Moreover, we give a non-trivial example and provide some applications of this inequality.


Introduction
A warped product N 1 × f N 2 of two Riemannian manifolds (N 1 , g 1 ) and (N 2 , g 2 ) is the product manifold N 1 × N 2 equipped with the warped product metric where f : N 1 → R + is a positive smooth function on N 1 . The function f is called the warping function [1,12]. When the warping function f is constant, M = N 1 × f N 2 is simply a Riemannian product. It is known that, for a vector field X on N 1 and a vector field Z on N 2 , we have formulas of Gauss and Weingarten are given respectively bỹ for X, Y ∈ Γ(T M ) and ξ ∈ Γ(T ⊥ M ), where D denotes the normal connection of the normal bundle T ⊥ M and A is the shape operator of M . The second fundamental form σ and the shape operator A of M are related by where , denotes the inner products on M andM with respect to their metrics. For any vector field X tangent to M , we put JX = P X + F X, (2.3) where P X is the tangential component and F X is the normal component of JX.
A pointwise slant submanifold is called a slant submanifold if its slant function θ is globally constant; such θ is called the slant angle of the slant submanifold (see [4,5]). Clearly, the same definitions apply to submanifolds in almost Hermitian manifolds. Obviously, complex and totally real submanifolds are slant submanifolds with slant angle θ = 0 and θ = π 2 , respectively, Definition 2.2. A pointwise slant submanifold M of a Kaehler manifoldM is called proper if its slant function θ satisfies 0 < θ < π 2 . Thus proper pointwise slant submanifolds are neither complex nor totally real.
From Lemma 2.1 of [13], we know that a submanifold M of an almost Hermitian manifoldM is a pointwise slant submanifold if and only if P 2 X = −(cos 2 θ)X, (2.5) for some real-valued function θ defined on M . The following relations are immediate consequences of (2.5): . The next relation for pointwise slant submanifolds of an almost Hermitian manifold follows easily from (2.1) and (2.5): for X ∈ Γ(T M ). (2.7) Then the complex Euclidean m-space C m = (E 2m , J) is a flat Kaehler manifold.

Pointwise CR-slant warped products: definition and example
Now, we provide the following definitions.
where D T is a complex distribution, D ⊥ is a totally real distribution, and D θ is a pointwise slant distribution whose slant function θ has values in (0, π 2 ). Denote by N T , N ⊥ , and N θ integrable submanifolds of D T , D ⊥ , and D θ , respectively. We call M a pointwise CR-slant warped product if the induced metric g on M is a warped product metric of the form where g B is the metric of B = N T × N ⊥ , g N θ is the metric on N θ , and f is a positive function depending only on B. This warped product is called proper if the warping function f is non-constant. In particular, if the slant function θ of D θ is a constant in (0, π 2 ), then M is called a CR-slant warped product. Notation 3.4. We simply denote the pointwise CR-slant warped product above by where µ is a J-invariant subbundle of the normal bundle T ⊥ M . We make the following definition.

By applying (3.4) and (3.6), it is easy to verify that the pointwise CR-slant warped
Proof. Using the polarization identity in Lemma 4.3 (ii) for vector fields X 3 , Y 3 ∈ Γ(D θ ), we find Thus we find from Lemma 4.3 (ii) and (4.4) that Now, replacing X 3 by P X 3 , we obtain Thus, if M is D ⊥ ⊕ D θ -mixed totally geodesic, then (4.5) yields the result. The following theorem gives a sharp inequality involving the norm σ of the second fundamental form for CR-slant warped products in any Kaehler manifold. Proof. From the definition of σ, we have where {e n+1 , . . . , e 2m−n } is a local orthonormal frame of the normal bundle. From (3.2), the above relation takes the form Leaving the last µ-components term in (5.2) and using the frame fields given above, we find