Counting geodesics on compact Lie groups (Q1694889)

This paper deals with counting the geodesics of a given length that connect two points. More precisely, given a tangent vector \(H\), the authors determine the number of connected components of the set \(\operatorname{fo}(H)\) (which contains the focal equivalents of \(H\)) and their dimensions. They consider the setting of a compact connected Lie group with a biinvariant metric and reduce the problem to the maximal torus by using the lattice, the diagram and the Weyl group to count the geodesics that lie outside the maximal torus. The obtained results are applied to recover in an elegant way characterizations of conjugate and cut points of compact semisimple Lie groups.