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Reverse Ricci-Curvature Bounds for Riemannian Submersions and Riemannian Maps

Ravindra Singh

Abstract

In this paper, we establish, for the first time, upper bounds of the Ricci--curvature for Riemannian submersions along the vertical distribution as well as along both the vertical and horizontal distributions. We derive their general forms and provide precise geometric characterisations of the equality cases. Furthermore, we obtain lower bounds of the Ricci--curvature for Riemannian maps, together with their general formulations and complete geometric characterisations of the equality cases. As applications, we apply these results to Riemannian submersions from real and complex space forms onto Riemannian manifolds, and to Riemannian maps from Riemannian manifolds into real and complex space forms.

Reverse Ricci-Curvature Bounds for Riemannian Submersions and Riemannian Maps

Abstract

In this paper, we establish, for the first time, upper bounds of the Ricci--curvature for Riemannian submersions along the vertical distribution as well as along both the vertical and horizontal distributions. We derive their general forms and provide precise geometric characterisations of the equality cases. Furthermore, we obtain lower bounds of the Ricci--curvature for Riemannian maps, together with their general formulations and complete geometric characterisations of the equality cases. As applications, we apply these results to Riemannian submersions from real and complex space forms onto Riemannian manifolds, and to Riemannian maps from Riemannian manifolds into real and complex space forms.
Paper Structure (10 sections, 12 theorems, 73 equations)

This paper contains 10 sections, 12 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. Background
  4. General Lower Bound for the Ricci Curvature Along the Vertical Distribution
  5. General Lower Bound for the Ricci Curvature Along the Vertical and Horizontal Distributions
  6. Applications
  7. Riemannian Maps
  8. Background
  9. General Lower Bound for the Ricci Curvature for Riemannian maps
  10. Applications

Key Result

Lemma 2.3

Hineva Suppose the $A=\left( a_{ij}\right)$ is any symmetric $\left( n\times n\right)$-matrix $\left( n\geq 2\right)$ such that ${\rm trace}\left( A\right) =a$ and the Frobenius norm $\left\Vert A\right\Vert =b$. Then The equality case in (eq-T-1) is true if and only if $A$ is of the form: where

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Proposition 5.1
  • Proposition 5.2
  • Theorem 5.3
  • ...and 11 more