Affine submanifolds and the theorem of Cartan-Ambrose-Hicks. (Q1872591)

In this article E. Cartan's theorem on the existence of submanifolds with prescribed tangent space is generalized to manifolds equipped with a linear connection. Generalizations of this type -- in the local and global context -- have been considered by \textit{N. Hicks} [Ill. J. Math. 10, 661--679 (1966; Zbl 0184.25203)]; but Hicks had to make additional unpleasant assumptions to be satisfied. The authors succeed in omitting these assumptions and are able to reprove the theorem of Cartan-Ambrose-Hicks as a special case of their results. Moreover, the results are for manifolds of different dimensions and arbitrary rank of the affine map.