Counting geodesics on compact symmetric spaces

TL;DR

This paper characterizes the inverse images of the exponential map on compact symmetric spaces as unions of focal orbits, linking their structure to algebraic data, and extends previous results on compact Lie groups.

Contribution

It provides a new description of geodesic inverse images in symmetric spaces using algebraic and topological data, extending prior work on compact Lie groups.

Findings

Describes inverse images as unions of focal orbits

Links orbit dimensions and components to algebraic data

Provides new proofs of known results

Abstract

We describe the inverse image of the Riemannian exponential map at a basepoint of a compact symmetric space as the disjoint union of so called focal orbits through a maximal torus. These are orbits of a subgroup of the isotropy group acting in the tangent space at the basepoint. We show how their dimensions (infinitesimal data) and connected components (topological data) are encoded in the diagram, multiplicities, Weyl group and lattice of the symmetric space. Obtaining this data is precisely what we mean by counting geodesics. This extends previous results on compact Lie groups. We apply our results to give short independent proofs of known results on compact symmetric spaces.