Isometric Immersions of Complete Riemannian Manifolds into Euclidean Space

Abstract

Let M be a complete Riemannian manifold of dimension n, with scalar curvature bounded from below. If the isometric immersion of M into euclidean space of dimension $n + q,q \leqslant n - 1$, is included in a ball of radius $\lambda$, then the sectional curvature K of M satisfies ${\lim \sup _M}K \geqslant {\lambda ^{ - 2}}$. The special case where M is compact is due to Jacobowitz.