Principal mappings of 3-dimensional Riemannian spaces into spaces of constant curvature

Abstract

As is well-known, the Gauss theorem, according to which any 2-dimensional Riemannian metric can be mapped locally conformally into an euclidean space, does not hold in three dimensions. We define in this paper transformations of a new type, that we call principal. They map 3-dimensional spaces into spaces of constant curvature. We give a few explicit examples of principal transformations and we prove, at the linear approximation, that any metric deviating not too much from the euclidean metric can be mapped by a principal transformation into the euclidean metric.