Eigenvalues and lambda constants on Riemannian submersions

TL;DR

This paper explores the relationship between eigenvalues and lambda constants in Riemannian submersions, analyzing how geometric properties transfer between base and total spaces, including warped products.

Contribution

It establishes new connections between lambda constants and eigenvalues on base and total spaces in Riemannian submersions, with detailed analysis on warped products.

Findings

Lambda constants relate to eigenvalues on base and total spaces

Eigenvalues on the total space can be estimated from the base

Warped products exhibit specific eigenvalue behaviors

Abstract

Given a Riemannian submersion, we study the relation between lambda constants introduced by G.Perelman on the base manifold and the total space of a Riemannian submersion. We also discuss the relationship between the first eigenvalues of Laplacians on the base manifold and that of the total space. The quantities on warped products are discussed in detail.