Classification of warped product pointwise semi-slant submanifolds in complex space forms

Abstract

The main principle of this paper is to show that, a warped product pointwise semi-slant
submanifold in a complex space form admitting shrinking or steady
gradient Ricci soliton, whose potential function is a well-defined warped function, is an Einstein warped
product pointwise semi-slant submanifold under extrinsic restrictions, and the second fundamental form
inequality with constant holomorphic curvature c, attaining the equality in [4]. Moreover, under some
geometric assumption, the connected and compactness with nonempty boundary are treated. In this case,
we propose a necessary and sufficient condition in terms of Dirichlet energy function which show that a
connected, compact warped product pointwise semi-slant submanifold of complex space forms must be a
Riemannian product. As more applications, for the first one, we prove that Mn is a trivial compact warped
product when the warping function exist the solution of PDE such as Euler-Lagrange equation. In the second
one, by imposing boundary conditions, we derive a necessary and sufficient condition in terms of Ricci
curvature and prove that a compact warped product pointwise semi-slant submanifold Mn of a complex
space form, is either a CR-warped product or just a usual Riemannian product manifold. We also discuss
some obstructions to these constructions in more details.