Affine Planar Geodesic Immersions with Maximal Codimension

This work gives a classification theorem for affine immersions with planar geodesics inthe case where the codimension is maximal. Vrancken classified parallel affine immersions in this caseand obtained, among others, generalized Veronese submanifolds. In this work it is shown that theimmersions with planar geodesics are the same as the parallel ones in the considered case. A geometricinterpretation of parallel immersions is also given: The affine immersions with pointwise planar normalsections (with respect to the equiaffine transversal bundle) are parallel. This result is verified forsurfaces in R~4 and for immersions with the maximal codimension.