Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Ravindra Singh

TL;DR

This work addresses the problem of relating intrinsic and extrinsic curvature in Riemannian submersions by proving a general Casorati inequality that couples the normalised scalar curvatures of the horizontal and vertical distributions with the normalised Casorati curvatures of both distributions. The main novelty is a general inequality for maps between arbitrary Riemannian manifolds, involving the delta-Casorati invariants, O'Neill tensors, and cross-terms arising from the ambient space, plus a complete equality characterisation. It then specializes the inequality to total spaces that are real, complex, generalized Sasakian, Sasakian, cosymplectic, Kenmotsu, and almost $C(\alpha)$-space forms, and extends to invariant, anti-invariant, slant, semi-slant, hemi-slant, and bi-slant submersions, with detailed examples. The results provide a unifying framework linking Casorati curvatures with scalar-curvature data across a wide class of geometric settings, offering tools for studying the geometry of submersions in both mathematical physics and differential geometry contexts.

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 $c(α)$-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.

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

TL;DR

This work addresses the problem of relating intrinsic and extrinsic curvature in Riemannian submersions by proving a general Casorati inequality that couples the normalised scalar curvatures of the horizontal and vertical distributions with the normalised Casorati curvatures of both distributions. The main novelty is a general inequality for maps between arbitrary Riemannian manifolds, involving the delta-Casorati invariants, O'Neill tensors, and cross-terms arising from the ambient space, plus a complete equality characterisation. It then specializes the inequality to total spaces that are real, complex, generalized Sasakian, Sasakian, cosymplectic, Kenmotsu, and almost -space forms, and extends to invariant, anti-invariant, slant, semi-slant, hemi-slant, and bi-slant submersions, with detailed examples. The results provide a unifying framework linking Casorati curvatures with scalar-curvature data across a wide class of geometric settings, offering tools for studying the geometry of submersions in both mathematical physics and differential geometry contexts.

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 -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.
Paper Structure (4 sections, 11 theorems, 117 equations, 1 table)

This paper contains 4 sections, 11 theorems, 117 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some Notations
  4. Casorati inequalities for both vertical and horizontal distributions of Riemannian submersions

Key Result

Lemma 6

Sahin_book Let $F:\left( {N_{1}}(c),g_{1}\right) \rightarrow \left( N_{2},g_{2}\right)$ be a Riemannian submersion from a complex space form onto a Riemannian manifold with slant angle $\theta$. Then the following statements are valid: for any $X,Y\in \ker F_{\ast }$. So, we obtain For more details about all cases of Riemannian submersion invariant, anti-invariant, slant, semi-slant, hemi-slant,

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Theorem 9
  • proof
  • ...and 15 more