First Chen Inequality for General Warped Product Submanifolds of a Riemannian Space Form and Applications

TL;DR

This paper establishes the first Chen inequality for warped product submanifolds in Riemannian space forms, linking intrinsic and extrinsic invariants, and applies it to minimal submanifold characterization and open problems.

Contribution

It proves the first Chen inequality for general warped product submanifolds in Riemannian space forms, connecting intrinsic and extrinsic invariants.

Findings

Derived a necessary condition for minimal submanifolds in Riemannian space forms.

Provided partial answers to longstanding problems by S.S. Chern.

Addressed open problems related to Chen inequalities and submanifold geometry.

Abstract

In this paper, the first Chen inequality is proved for general warped product submanifolds in Riemannian space forms, this inequality involves intrinsic invariants ($\delta$-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.