# Notes on flat pseudo-Riemannian manifolds

## TL;DR

This paper surveys affine and pseudo-Riemannian geometry, characterizing flat manifolds, Lie groups with flat metrics, and exploring their properties, including completeness and unimodularity, with a focus on Lie groups and hyperbolic metrics.

## Contribution

It provides new characterizations of flat pseudo-Riemannian manifolds and Lie groups, and explores conditions for flatness, completeness, and unimodularity in this context.

## Key findings

- No connected semisimple Lie group admits a flat affine connection.
- Flat pseudo-Riemannian Lie groups are characterized by specific properties.
- Completeness of Levi-Civita connection is equivalent to unimodularity for flat pseudo-metrics.

## Abstract

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.