On the General Problem of Structure-Preserving Submersions

TL;DR

This paper develops a comprehensive geometric framework for structure-preserving submersions, applying it to generalize classical theorems like Herglotz-Noether across various dimensions and conformally-flat spaces.

Contribution

It introduces a novel geometric approach to structure-preserving submersions and extends the Herglotz-Noether theorem to broader contexts.

Findings

Generalized Herglotz-Noether theorem to arbitrary dimensions

Extended classical results to conformally-flat ambient spaces

Provided a unified geometric framework for physics-related problems

Abstract

In this paper we give a general geometrical framework for working with problems that can be described as a structure-preserving submersion defined on a suitable space with a geometrical structure. We give many examples of how to formulate familiar problems arsing from physics in our framework. Then, as an application of this framework we derive the new results of generalisations of the classical Herglotz-Noether theorem to arbitrary dimensions and to all conformally-flat ambient spaces. Possible further directions for research are also discussed.