An embedding problem

Abstract

This paper contains two results. First it is shown that the three-dimensional Riemannian space, which is invariant under the transformations of the rotation group, cannot be embedded in a four-dimensional Euclidean space (except, of course, for the three-dimensional sphere). Second, the one parametric family of three-spaces with the above symmetry, which can be embedded in a four-dimensional unit sphere, is found and the embedding is constructed.