On curvature homogeneous and locally homogeneous affine connections

Abstract

This paper deals with curvature homogeneous affine connections on 2-dimensional manifolds. We give a sufficient condition for a protectively flat curvature homogeneous connection to be locally homogeneous and show how to construct curvature homogeneous connections that are not locally homogeneous. The concept of a curvature homogeneous Riemannian connection was introduced by I. M. Singer in [5] and then studied in many papers. A similar notion can be defined in affine geometry. Curvature homogeneous affine connections do not have, in general, properties similar to Riemannian ones. For instance, a curvature homogeneous Riemannian connection on a 2-dimensional manifold is automatically locally symmetric and consequently locally homogeneous. As regards the affine case, it is easy to find curvature homogeneous affine connections on 2-dimensional manifolds which are not locally homogeneous. For instance, connections with symmetric Ricci tensor of rank 1 on connected manifolds are always curvature homogeneous and only in exceptional cases locally homogeneous. In this note we begin to study curvature homogeneous affine connections on 2-dimensional manifolds. The differences between the curvature homogeneity, locally homogeneity and local symmetry conditions are illustrated by means of protectively flat connections, which, in some sense, are the closest to locally symmetric ones. 1 Manifolds are assumed to be connected and connections torsion-free. For a given connection its curvature and Ricci tensors will be denoted by R and Ric respectively. Definition 1.1. Let V be a connection on a manifold M. It is called curvature homogeneous of order r if and only if for every x, y E M there exists a linear isomorphism F: TsM TyM such that F*(ViR)y = (V'R), for every i = 0, ..., r. By a curvature homogeneous connection we mean a connection which is curvature homogeneous of order 0. Received by the editors November 15, 1994. 1991 Mathematics Subject Classification. Primary 53B05, 53C30.