Some Theorems on the Isometric Imbedding of Compact Riemann Manifolds in Euclidean Space

Abstract

This paper will be concerned with some estimates on the lower bound of the dimension of the Euclidean space in which a compact Riemann manifold can be imbedded isometrically, if its curvatures satisfy certain conditions. Our basic geometrical idea is a very simple one. Denote by M a compact Riemann manifold of dimension n in an Euclidean space E of dimension n + N, the Riemann metric on M being induced by the imbedding. Let be a fixed point of E. The distance OP, P e M, is a continuous function in M and attains a maximum at a point P0 e M, since M is compact. It is intuitively clear that M will be concave toward 0 at Po, so that there will be some restrictions on the Riemann curvature of M at P0. If M is given abstractly, the imbedding will not be possible, if these restrictions are not fulfilled by the given Riemann metric at any of the points of M. Actually, however, if the difference N of the dimensions of M and E is greater than one, the implication of this geometrical fact on the Riemann curvature of M is not very simple. The question leads to algebraic problems which probably do not have simple answers. We propose to give in this paper a few conclusions which can be drawn. It should be mentioned that the above geometrical idea has been used by Tompkins1 to prove that a locally flat compact Riemann manifold of dimension n cannot be isometrically imbedded in an Euclidean space of dimension 2n 1. Among other things this theorem will be generalized and the invariants entering in the problem geometrically interpreted. As for the differentiability assumptions we suppose our manifold and the imbedding to be of class > 4.