Locally Homogeneous Triples: Extension Theorems for Parallel Sections and Parallel Bundle Isomorphisms

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 \((g, P\mathop {\rightarrow }\limits ^{p} M,A)\), where \(p:P\rightarrow M\) 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 \(\varphi :U\rightarrow U'\) between open neighbourhoods \(U\ni x\), \(U'\ni x'\) with \(\varphi (x)=x'\), and a \(\varphi \)-covering bundle isomorphism \(\Phi :P_U\rightarrow P_{U'}\) such that \(\Phi ^*(A_{U'})=A_U\). If \((g,P\mathop {\rightarrow }\limits ^{p} M,A)\) 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 to describe explicitly moduli spaces of locally homogeneous triples on Riemann surfaces.