2017/07/21 by Alberto Abad Medina, Medina, Alberto, Omar Saldarriaga +3
Mathematics · #17D25 (Secondary) #53C07 #54H15 (Primary) #57S20 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1707.07030
openalex publication_date 2017/07/21 · openalex created_date 2017/07/31 · openalex updated_date 2026/07/28
This paper deals with affine connections on real manifolds. We give a new characterization of flat affine connections on real manifolds by means of certain affine representations of the Lie group of automorphisms preserving the connection. Then we specialize the characterization to the case of a left invariant connection on a Lie group. In the last case, we show the existence of a Lie group endowed with a flat affine bi-invariant connection whose Lie algebra contains the Lie algebra of complete infinitesimal affine transformations of the given Lie group. We also prove some results about flat affine manifolds whose group of diffeomorphisms admit a flat affine bi-invariant structure. The paper is illustrated with several examples.