2016/05/02 by Arash Bazdar, Bazdar, Arash
Mathematics · #53C05 #53C30 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C05 #msc:53C30
paper · pdf · doi:10.48550/arxiv.1605.00610
LaTeX, 15 pages
arxiv created 2017/02/13 · arxiv updated 2017/02/14
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\stackrelp→ M,A), where p:P→ 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'∈ M there exists an isometry φ:U→ U' between open neighborhoods U\ni x, U'\ni x' with φ(x)=x', and a φ-covering bundle isomorphism Φ:PU→ PU' such that Φ^*(AU')=AU. If (g,P\stackrelp→ 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.