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 $(g,P\stackrel{p}{\to }M,A)$, 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 $x$, $x^{\prime }\in M$ there exists an isometry $\phi :U\to U^{\prime }$ between open neighborhoods $U\ni x$, $U^{\prime }\ni x^{\prime }$ with $\phi (x)=x^{\prime }$, and a $\phi$-covering bundle isomorphism $\Phi :P_U\to P_{U^{\prime }}$ such that ${\Phi }^{\ast }(A_{U^{\prime }})=A_U$. 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.

arXiv commentLaTeX, 15 pages