Knots in Riemannian manifolds

TL;DR

This paper investigates the topology of knots in positively curved Riemannian spheres, proving that certain totally geodesic submanifolds are topologically spheres with knot complements having fundamental group isomorphic to integers.

Contribution

It establishes a topological classification of knots in spheres with positive curvature, showing that specific totally geodesic submanifolds are homeomorphic to spheres and their complements have fundamental group Z.

Findings

Totally geodesic submanifolds in S^n are homeomorphic to S^{n-2}.

Knot complements in these spheres have fundamental group isomorphic to Z.

Results apply for spheres with dimension n ≥ 5.

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 $\pi_1(S^n-K)\cong \Bbb Z$.