Geometry of pointwise CR-Slant warped products in Kaehler manifolds

Abstract

 We call a submanifold $M$ of a Kaehler manifold $\tilde{M}$ a pointwise CR-slant warped product if it is a warped product, $B\times_{\!f} N_\theta$, of a CR-product $B=N_T\times N_\perp$ and a proper pointwise slant submanifold $N_\theta$ with slant function $\theta$, where $N_T$ and $N_\perp$ are complex and totally real submanifolds of $\tilde{M}$. We prove that if a pointwise CR-slant warped product $B\times_{\!f} N_\theta$ with $B=N_T\times N_\perp$ in a Kaehler manifold is weakly ${\mathfrak{D}^\theta}$-totally geodesic, then it satisfies
\[
\|\sigma\|^2\geq 4s\left\{ (\csc^2\theta+\cot^2\theta)\|\nabla^T(\ln f)\|^2 +
(\cot^2\theta)\|\nabla^\bot(\ln f)\|^2 \right\},
\]
where $N_T$, $N_\perp$, and $N_\theta$ are complex, totally real and proper pointwise slant submanifolds of $\tilde{M}$, respectively, and $s=\frac{1}{2}\dim N_\theta$. 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.