B.-Y. Chen's inequalities for Riemannian submersion and their applications

TL;DR

This paper introduces B.-Y. Chen inequalities for Riemannian submersions, relating intrinsic and extrinsic invariants, with applications to specific geometric structures and illustrative examples.

Contribution

It establishes new inequalities for Riemannian submersions and explores their equality cases, extending to various geometric contexts.

Findings

Derived inequalities for vertical, horizontal, and mixed distributions.

Identified conditions for equality cases in these inequalities.

Provided examples illustrating both equality and inequality cases.

Abstract

In this paper, we introduce B.-Y. Chen inequalities for Riemannian submersions between Riemannian manifolds. We derive these inequalities for vertical, horizontal, and mixed distributions, establishing relationships between intrinsic invariants and extrinsic invariants. We also investigate the corresponding equality cases. As applications, the results are obtained for submersions whose total space is a real, complex, generalized Sasakian space form. Several examples are provided to illustrate both equality and strict inequality cases.