  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.
