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

摘要

Let $M$ be a differentiable manifold and $K$ a Lie group. A locallyhomogeneous 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 thefollowing 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 bundleisomorphism $\Phi:P_U\to P_{U'}$ such that $\Phi^*(A_{U'})=A_U$. If$(g,P\stackrel{p}{\to} M,A)$ is a locally homogeneous triple on $M$, one canendow the total space $P$ with a locally homogeneous Riemannian metric suchthat $p$ becomes a Riemannian submersion and $K$ acts by isometries. Thereforethe classification of locally homogeneous triples on a given manifold $M$ is animportant problem: it gives an interesting class of geometric manifolds whichare fibre bundles over $M$. In this article we will prove a classification theorem for locallyhomogeneous triples. We will use this result in a future article in order todescribe explicitly moduli spaces of locally homogeneous triples on Riemannsurfaces.