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 of type M-n = N-T(n1) x(f) N-theta(n2) in a complex space form (M) over tilde (2m)(c) admitting shrinking or steady gradient Ricci soliton, whose potential function is a well-define warped function, is an Einstein warped product pointwise semi-slant submanifold under extrinsic restrictions on the second fundamental form inequality 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 M-n 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 M-n 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.