On some spaces of minimal geodesics in Riemannian symmetric spaces (Q2907042)

Let \((P,o)\) be a pointed connected irreducible symmetric space of compact type. The authors study the space \(\Lambda ^{\text{min}}(P;o,p)\) of all shortest geodesics from \(o\) to \(p\) in \(P\). This space is an important ingredient in \textit{R. Bott}'s [Ann. Math. (2) 70, 313--337 (1959; Zbl 0129.15601)] proof of his periodicity theorem. He showed that this space is symmetric by showing that the orbits under the action of the group of isometries fixing both \(o\) and \(p\) are symmetric.NEWLINENEWLINEThe authors refine this result when \(p\) lies in the center of \((P,o)\). Specifically, they give a complete description of geodesics in these orbits in terms of extrinsically symmetric elements, defined as \(\xi\in T_oP\) such that \(\xi\neq 0\) and \((\text{ad}_\xi)^3=-\text{ad}_\xi\). They also show that each connected component of the set of midpoints of the geodesics in \(\Lambda ^{\text{min}}(P;o,p)\) is a symmetric \(R\)-space which is totally geodesically and isometrically embedded in \(P\).