General Casorati Inequality for Riemannian Submersions Involving Horizontal and Vertical Casorati Curvatures and Applications

TL;DR

This paper establishes a new Casorati inequality for Riemannian submersions, relating horizontal and vertical curvatures, with applications across various geometric structures and explicit equality cases.

Contribution

It introduces a comprehensive Casorati inequality for Riemannian submersions, characterizes equality cases, and applies the results to multiple geometric contexts and submersion types.

Findings

Derived a general Casorati inequality for Riemannian submersions.

Characterized the equality cases completely.

Applied inequalities to various geometric structures and submersion types.

Abstract

In this paper, we develop and introduce a Casorati inequality for Riemannian submersions involving the Casorati curvatures of both the vertical and horizontal distributions. A general form of the inequality is derived for Riemannian submersions between Riemannian manifolds, and the corresponding equality cases are completely characterised. As applications, we obtain the inequality for Riemannian submersions whose total spaces are real, complex, generalised Sasakian, Sasakian, cosymplectic, Kenmotsu, and almost $c(\alpha)$-space forms. For each theorem, we present illustrative examples. Some of these examples achieve equality, while others do not. Furthermore, these inequalities are derived for invariant, anti-invariant, slant, semi-slant, hemi-slant, and bi-slant Riemannian submersions.