On invariant projectively flat affine connections
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.
left-invariant projectively flat connection
scientific article; zbMATH DE number 1331850