On the realizability of projectively flat connections on surfaces

Abstract

The main purpose of this note is to explain the nature of an example from [4] showing that, in contrast with locally symmetric connections on 2-dimensional manifolds, projectively flat connections on such manifolds are not necessarily realizable on nondegenerate surfaces in \( \bf R^3 \). The connection given in [4] has symmetric Ricci tensor of rank 1. In the present note we give a necessary condition for the realizability of such connections on nondegenerate surfaces in \( \bf R^3 \) (in particular, by Blaschke immersions). It allows to produce many non-realizable connections. We also give a new example of a class of projectively flat connections which have nondegenerate Ricci tensor and are realizable as Blaschke ones in a unique way.