Knots in Riemannian manifolds

Abstract

In this paper we study submanifold with nonpositive extrinsic curvature in a positively curved manifold. Among other things we prove that, if $K\subset (S^n, g)$ is a totally geodesic submanifold in a Riemannian sphere with positive sectional curvature where $n\ge 5$, then $K$ is homeomorphic to $S^{n-2}$ and the fundamental group of the knot complement $\pi_1(S^n-K)\cong \Bbb Z$.