Graphs and their parallel groups

Abstract

Abstract Given an immersion of a manifoldf: M?R n+k , dimensionM=n, the parallel groupP(f) off is formed by the diffeomorphisms ofM such that the normalk-planes at points of each orbit are parallel. In [3] we studied the parallel group of a plane closed curve. Here we concentrate on immersionsf: R n ?R n+1, special attention being paid to graphs of smooth maps fromR toR. Graphs of smooth mapsf: S n ?R m are also dealt with and we characterise those maps of which the graph has nontrivial parallel group. To end up we find a sufficient condition for the triviality of the tangent group.