Submanifolds with Nonpositive Extrinsic Curvature

Abstract

The results of this chapter show that isometric immersions \(f\colon M^n\to \tilde {M}^m\) with low codimension and nonpositive extrinsic curvature at any point must satisfy strong geometric conditions. That f has nonpositive extrinsic curvature at any point means that the sectional curvature KM(σ) of Mn along any plane σ does not exceed the corresponding sectional curvature \(K_{\tilde M}(f_*\sigma )\) of \(\tilde {M}^m\). The simplest result along this line is that a two-dimensional surface with nonpositive curvature in \(\mathbb {R}^3\) cannot be compact. This is a consequence of the fact that at a point of maximum of a distance function on a compact surface in \(\mathbb {R}^3\) the Gaussian curvature must be positive. It turns out that the simple idea in the proof of this elementary fact has far-reaching generalizations for non-necessarily compact submanifolds in fairly general ambient Riemannian manifolds.