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General Chen-Ricci inequalities for Riemannian submersions and Riemannian maps

Ravindra Singh, Kiran Meena, Kapish Chand Meena

Abstract

In this paper, we derive general forms of the Chen-Ricci inequalities for Riemannian submersions between Riemannian manifolds. We also derive general forms of the Chen-Ricci and improved Chen-Ricci inequalities for Riemannian maps between Riemannian manifolds, involving relations between the curvatures of subspaces of the source and target spaces. Further, we illustrate equality cases for all these general forms with two examples. These general forms yield new, easy, and elegant techniques that are fruitful in obtaining the Chen-Ricci inequalities for such smooth mappings with various structured manifolds. As applications, utilizing these general forms, we explicitly establish Chen-Ricci inequalities when the source manifolds of Riemannian submersions and the target manifolds of Riemannian maps belong to broader classes, such as generalized complex and generalized Sasakian space forms, particularly including real, complex, real Kähler, Sasakian, Kenmotsu, cosymplectic, and almost $C(α)$ space forms. We also validate our approach by imposing appropriate conditions toward various particular existing cases.

General Chen-Ricci inequalities for Riemannian submersions and Riemannian maps

Abstract

In this paper, we derive general forms of the Chen-Ricci inequalities for Riemannian submersions between Riemannian manifolds. We also derive general forms of the Chen-Ricci and improved Chen-Ricci inequalities for Riemannian maps between Riemannian manifolds, involving relations between the curvatures of subspaces of the source and target spaces. Further, we illustrate equality cases for all these general forms with two examples. These general forms yield new, easy, and elegant techniques that are fruitful in obtaining the Chen-Ricci inequalities for such smooth mappings with various structured manifolds. As applications, utilizing these general forms, we explicitly establish Chen-Ricci inequalities when the source manifolds of Riemannian submersions and the target manifolds of Riemannian maps belong to broader classes, such as generalized complex and generalized Sasakian space forms, particularly including real, complex, real Kähler, Sasakian, Kenmotsu, cosymplectic, and almost space forms. We also validate our approach by imposing appropriate conditions toward various particular existing cases.

Paper Structure

This paper contains 14 sections, 6 theorems, 89 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Generalized space forms
  3. General Chen-Ricci inequalities for Riemannian submersions with applications
  4. Background
  5. General Chen-Ricci inequalities
  6. Example
  7. Applications
  8. General Chen-Ricci inequalities for Riemannian maps with applications
  9. Background
  10. General Chen-Ricci inequality
  11. Improved general Chen-Ricci inequality
  12. Example
  13. Applications
  14. Conclusion with more applications and validations

Key Result

Theorem 3.1

Let $F:(M_{1}^{m_{1}},g_{1}) \to (M_{2}^{m_{2}},g_{2})$ be a Riemannian submersion between two Riemannian manifolds. Then and Further, equality cases are as follows.

Theorems & Definitions (15)

  • Theorem 3.1
  • proof
  • Example 3.2
  • Theorem 3.3
  • proof : Proof of $(i)$
  • proof : Proof of $(ii)$
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 5 more