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Geometric Properties and Spectral Estimates on Warped Products

Josué Meléndez, Eduardo Rodríguez-Romero, Jonatán Torres Orozco

Abstract

We establish an integral inequality for the Ricci curvature of a certain class of warped products $M\times_fN$, where the equality holds if and only if it is simply a Riemannian product. We also give a sufficient condition for the intersection of a warped product $M=\mathbb{R}\times_fP$ with a totally geodesic hypersurface $N$ in an arbitrary Riemannian space to be a totally geodesic slice of $M$. In addition, we establish some spectral estimates for the Laplacian of a submanifold $N$ that intersects a warped product in the same ambient manifold.

Geometric Properties and Spectral Estimates on Warped Products

Abstract

We establish an integral inequality for the Ricci curvature of a certain class of warped products , where the equality holds if and only if it is simply a Riemannian product. We also give a sufficient condition for the intersection of a warped product with a totally geodesic hypersurface in an arbitrary Riemannian space to be a totally geodesic slice of . In addition, we establish some spectral estimates for the Laplacian of a submanifold that intersects a warped product in the same ambient manifold.

Paper Structure

This paper contains 7 sections, 13 theorems, 95 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Preliminaries on warped products and intersections
  3. Proofs of the main results
  4. Proof of Theorem \ref{['theo:integral-inequality']}
  5. Proof of Theorem \ref{['theo:normality']}
  6. Proof of Theorem \ref{['theo:firsteigen']}
  7. Proof of Theorem \ref{['theo: Reilly']}

Key Result

Theorem 1

Let $(M,g_M)$ and $(N,g_N)$ be Riemannian manifolds with $\dim M=m$, and assume that $M$ is Ricci flat. Let ${ \macc@depth1 \frozen@everymath{\mathgroup\macc@group} \macc@set@skewchar \macc@nested@a111{M} }=M\times_f N$ be a compact warped product with warping function $f$. Then where $\operatorname{Ric}$ denotes the Ricci curvature of ${ \macc@depth1 \frozen@everymath{\mathgroup\macc@gr

Figures (1)

  • Figure 1: A rotation hypersurface $M$ normal to the parallel plane $x_1=t_0$

Theorems & Definitions (29)

  • Theorem 1
  • Corollary 1
  • Example 1
  • Example 2
  • Example 3
  • Definition 1
  • Definition 2
  • Remark 1
  • Theorem 2
  • Remark 2
  • ...and 19 more