Published Paper
Inserted: 15 apr 2022
Last Updated: 23 sep 2023
Journal: Adv. Pure Appl. Math.
Volume: 14
Number: 3
Pages: 22-40
Year: 2023
Abstract:
In this paper, the first Chen inequality is proved for general warped product submanifolds in Riemannian space forms, this inequality involves intrinsic invariants (δ-invariant and sectional curvature) controlled by an extrinsic one (the mean curvature vector), which provides an answer for Problem 1. As a geometric application, this inequality is applied to derive a necessary condition for the immersed submanifold to be minimal in Riemannian space forms, which presents a partial answer for the well-known problem proposed by S.S. Chern, Problem 2. For further research directions, we address a couple of open problems; namely Problem 3 and Problem 4.
Keywords: δ-invariant; warped product; minimal submanifolds; Riemannian space forms