Focal radii of orbits

TL;DR

This paper proves that for any effective action of a compact Lie group on a sphere, there exists an orbit with principal curvatures bounded above by a specific constant, providing explicit geometric bounds.

Contribution

It establishes an explicit upper bound on the principal curvatures of some orbit in any effective compact Lie group action on a sphere.

Findings

Existence of an orbit with bounded principal curvatures

Explicit bound of 4√14 on principal curvatures

Applicable to all effective actions of compact Lie groups

Abstract

We show that every effective action of a compact Lie group $K$ on a unit sphere $S^n$ admits an explicit orbit whose principal curvatures are bounded from above by $4\sqrt{14}$.