PRINCIPAL MAPPINGS OF 3-DIMENSIONAL RIEMANNIAN SPACES INTO SPACES OF CONSTANT CURVATURE

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