A Contribution to Differential Geometry in the Large

Abstract

The contribution can be simply formulated in the following theorem. I first formulate HYPOTHESIS A: Let Mn n > 2, be a manifold of class C'(3 ? a? w) with a Riemannian metric given by a positive-definite tensor gij(i, j = 1, ***, n) of class Ca such that M' is complete with respect to the metric. THEOREM 1. Assume A. If the Riemannian curvature K(P, -y) (P e yf, y a 2-dimensional plane through P) formed from gij satisfies