On graphs in Euclidean spaces with minimal total curvature

Abstract

In this paper we study the flat maps, that is, the polygonal maps with minimal total curvature, from a finite graph G to a Euclidean space E n that were recently defined and studied by K. Taniyama. We investigate the local behavior of these flat maps. As a consequence we determine the vertex dimension and the curvature dimension, which are invariants of graphs included by flat maps, of complete graphs.