The concept of Cheeger deformations on fiber bundles with compact structure group

TL;DR

This paper introduces Cheeger deformations on fiber bundles with compact structure groups, unifies existing results, and demonstrates their power through simplified proofs and extensions, serving as a survey and standardization of the technique.

Contribution

It systematically defines Cheeger deformations on fiber bundles and re-proves and extends several key results, unifying them in a single framework.

Findings

Re-proved Schwachhöfer–Tuschmann Theorem on bi-quotients

Extended work on regularization properties of Cheeger deformations

Unified presentation of known results using a common language

Abstract

The purpose of this paper is two-fold: we systematically introduce the notion of Cheeger deformations on fiber bundles with compact structure groups, and recover in a very simple and unified fashion several results that either already appear in the literature or are known by experts, though are not explicitly written elsewhere. We re-prove: Schwachh\"ofer--Tuschmann Theorem on bi-quotients, many results due to Fukaya and Yamaguchi, as well as, naturally extend the work of Searle--Sol\'orzano--Wilhelm on regularization properties of Cheeger deformations, among others. In this sense, this paper should be understood as a survey intended to demonstrate the power of Cheeger deformations. Even though some of the results here appearing may not be known as stated in the presented form, they were already expected, being our contribution to the standardization and spread of the technique via a unique language.