2019/03/21 by Fabricio Valencia, Valencia, Fabricio
Mathematics · Physics and Astronomy · #22E60 #52C20 #53A15 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1903.08940
openalex publication_date 2019/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In these notes we survey basic concepts of affine geometry and their interaction with Riemannian geometry. We give a characterization of affine manifolds which has as counterpart those pseudo-Riemannian manifolds whose Levi-Civita connection is flat. We show that no connected semisimple Lie group admits a left invariant flat affine connection. We characterize flat pseudo-Riemannian Lie groups. For a flat left-invariant pseudo-metric on a Lie group, we show the equivalence between the completeness of the Levi-Civita connection and unimodularity of the group. We emphasize the case of flat left invariant hyperbolic metrics on the cotangent bundle of a simply connected flat affine Lie group. We also discuss Lie groups with bi-invariant pseudo-metrics and the construction of orthogonal Lie algebras.