Locally homogeneous triples. Extension theorems for parallel sections and parallel bundle isomorphisms

TL;DR

This paper introduces a classification theorem for locally homogeneous triples consisting of a Riemannian metric, principal bundle, and connection on a manifold, with applications to understanding moduli spaces on Riemann surfaces.

Contribution

It provides a new classification framework for locally homogeneous triples, linking local symmetries to global geometric structures and bundle isomorphisms.

Findings

Classification theorem for locally homogeneous triples established

Construction of locally homogeneous Riemannian metrics on bundle total spaces

Application to moduli space descriptions on Riemann surfaces

Abstract

Let $M$ be a differentiable manifold and $K$ a Lie group. A locally homogeneous triple with structure group $K$ on $M$ is a triple $(g, P\stackrel{p}{\to} M,A)$, where $p:P\to M$ is a principal $K$-bundle on $M$, $g$ is Riemannian metric on $M$, and $A$ is connection on $P$ such that the following locally homogeneity condition is satisfied: for every two points $x$, $x'\in M$ there exists an isometry $\varphi:U\to U'$ between open neighborhoods $U\ni x$, $U'\ni x'$ with $\varphi(x)=x'$, and a $\varphi$-covering bundle isomorphism $\Phi:P_U\to P_{U'}$ such that $\Phi^*(A_{U'})=A_U$. If $(g,P\stackrel{p}{\to} M,A)$ is a locally homogeneous triple on $M$, one can endow the total space $P$ with a locally homogeneous Riemannian metric such that $p$ becomes a Riemannian submersion and $K$ acts by isometries. Therefore the classification of locally homogeneous triples on a given manifold $M$ is an important problem: it gives an interesting class of geometric manifolds which are fibre bundles over $M$. In this article we will prove a classification theorem for locally homogeneous triples. We will use this result in a future article in order to describe explicitly moduli spaces of locally homogeneous triples on Riemann surfaces.