Focal radii of orbits

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 √ 14.

Highly curved orbit spaces

“…First, the more extrinsically curved a G-orbit is, the closer its focal points are, and thus the sooner a normal geodesic starting there ceases to be minimizing. It is shown in [GS18] that the infimum over all actions (coming from irreducible Date: September 11, 2018. representations, non-transitive on the unit sphere) of the supremum over all orbits of their focal radii is bounded away from zero. Second, and more relevant to this paper, the Bonnet-Myers argument implies that diam X ≤ π/ √ κ X .…”