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General Chen's First Inequalities for Riemannian Submersions and Their Applications

Ravindra Singh

Abstract

In this paper, we introduce and develop the concepts of Chen's first inequalities for Riemannian submersions between Riemannian manifolds. We derive general forms of Chen's first inequalities and analyse their corresponding equality cases. As applications, we apply them to various Riemannian submersions whose total space is real, complex, generalised Sasakian, Sasakian, Kenmotsu, cosymplectic, and $C(α)$ space forms. We construct examples that satisfy the assumptions of the theorems; we observe that equality holds in some examples, while in others it does not.

General Chen's First Inequalities for Riemannian Submersions and Their Applications

Abstract

In this paper, we introduce and develop the concepts of Chen's first inequalities for Riemannian submersions between Riemannian manifolds. We derive general forms of Chen's first inequalities and analyse their corresponding equality cases. As applications, we apply them to various Riemannian submersions whose total space is real, complex, generalised Sasakian, Sasakian, Kenmotsu, cosymplectic, and space forms. We construct examples that satisfy the assumptions of the theorems; we observe that equality holds in some examples, while in others it does not.
Paper Structure (7 sections, 11 theorems, 104 equations, 1 table)

This paper contains 7 sections, 11 theorems, 104 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. O'Neill tensors
  4. Relations between Riemannian curvature tensors
  5. $\delta \left( 2\right)$-invariant
  6. Chen's first inequality along vertical distributions
  7. Chen's first inequality along Horizontal and Vertical distributions

Key Result

Lemma 2.4

Let $F:(M_{1},g_{1})\rightarrow (M_{2},g_{2})$ be a Riemannian submersion between Riemannian manifolds with $\dim M_{1}=n$ and $\dim M_{2}=m$. If the fiber dimension is $r=\dim {\mathcal{V}}_{p}>2$, then for $2$-plane $\Pi \subset {\mathcal{V}}_{p}$ and $2$-plane ${\Bbb P}\subset {\mathcal{H}}_{p}$, where From Chen_2000 Similarly, we define where and where

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Example 3.3
  • ...and 24 more