Affine submanifolds and the Theorem of Cartan-Ambrose-Hicks

Abstract

In this article E. Cartan's theorem about the existence of (local) totally geodesic submanifolds with a prescribed tangent plane is generalized to manifolds which are equipped only with a linear connection; there is also given a global version of the theorem. The results are used for a short and geometric proof of the Theorem of Cartan-Ambrose-Hicks, and more generally of a generalization which is concerned with the existence of affine maps of arbitrary rank.