ON ISOMETRIC IMMERSIONS OF A RIEMANNIAN SPACE WITH LITTLE REGULARITY

Abstract

We consider a Riemannian metric in an open subset of the d-dimensional Euclidean space and assume that its Riemann curvature tensor vanishes. If the metric is of class C2, a classical theorem in differential geometry asserts that the Riemannian space is locally isometrically immersed in the d-dimensional Euclidean space. We establish that if the metric belongs to the Sobolev space W1,∞ and its Riemann curvature tensor vanishes in the space of distributions, then the Riemannian space is still locally isometrically immersed in the d-dimensional Euclidean space.