Knots in Riemannian manifolds

Abstract

In this paper we study submanifolds 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 of codimension 2 in a Riemannian sphere with positive sectional curvature where n ≥ 5, then K is homeomorphic to S n–2 and the fundamental group of the knot complement \({\pi _1(S^n-K)\cong \mathbb{Z}}\).