Geometry of Cyclic Quotients, I: Knotted Totally Geodesic Submanifolds in Positively Curved Spheres

TL;DR

This paper constructs specific positively curved metrics on spheres that admit knotted geodesics and totally geodesic submanifolds, using techniques inspired by negatively curved manifold constructions.

Contribution

It demonstrates the existence of positive curvature metrics on spheres with knotted geodesics and special submanifolds, introducing new geometric constructions.

Findings

Existence of positive curvature metric with a torus knot as a geodesic

Construction of a metric with a totally geodesic projective plane of Euler number four

Use of techniques borrowed from negatively curved manifold theory

Abstract

We prove that there exists a metric of positive curvature in a three-sphere which admits a given torus knot as a closed geodesic.We also sketch a construction of a metric in a four sphere, very likely of positive curvature, which admits a totally geodesic projective plane with Euler number four. Surpisingly, the technique borrows a lot from the Mostow-Siu-Gromov-Thurston constuction of exotic negatively curved manifolds.