On invariant projectively flat affine connections
type: Item
instance of: scholarly article
title: On invariant projectively flat affine connections
zbMATH Open document ID: 0978.53097
author: Hajime Urakawa
published in: Hokkaido Mathematical Journal
publication date: 2002-01-25T00:00:00Z
review text: In this paper projectively flat affine connections are studied. An affine connection on a smooth manifold is called flat if its curvature tensor vanishes. A torsion-free and Ricci-symmetric affine connection is called projectively flat if it is locally projectively equivalent to a flat affine connection. In the theory of compact manifolds with a geometric structure, flat manifolds and their fundamental groups have been studied by Milnor, Auslander and others. In this context the question arises to determine the Lie groups admitting a left-invariant flat torsion-free affine connection. A real simple Lie group does not admit such a connection, whereas the existence question for solvable Lie groups is open in general. The situation is different for invariant projectively flat connections. The first result of this paper is a classification of all irreducible Riemannian symmetric spaces admitting an invariant projectively flat connection. The second result is a partial classification of real simple Lie groups admitting a left-invariant projectively flat connection. Let \(G\) be a real simple Lie group with Lie algebra \(\text{lie} g\). If \(G\) admits an invariant projectively flat connection then the Lie algebra \(\text{lie} g\oplus \mathbb R\) admits a certain linear representation of dimension \(\dim G +1\). This necessary condition can be easily checked by using a dimension argument. The condition is not sufficient in general. The author obtains a list of groups for which such a representation exists. For the first part of the list, that is, for the groups \(SL_n(\mathbb R)\) and \(SU^*(2n)\), he shows that there exists a left-invariant projectively flat connection. For the second part (\(SU(r,s)\) and \(O(r,s)\) for certain \(r,s\)), the question remains open.
reviewed by: Dietrich Burde
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 53C05
zbMATH DE Number: 1331850
zbMATH Keywords: projectively flat affine connections
zbMATH Keywords: irreducible Riemannian symmetric spaces
zbMATH Keywords: left-invariant projectively flat connection
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1351001217
OpenAlex ID: W2056880414
Wikidata QID: Q125379747
DOI: 10.14492/HOKMJ/1351001217
description: scientific article; zbMATH DE number 1331850