On Some Spaces of Minimal Geodesics in Riemannian Symmetric Spaces

Centrioles in symmetric spaces

“…If X ∈ a + is an element of type I and if Y ∈ a + is an element of type II, III, or IV, then the corresponding centrioles K. (exp((1/2) Remark 16. Both [MQ1,Lemma 4.4] and our Theorem 12 imply that the number of poles of (P , o) coincides with the number of different elements of type I in the closed fundamental Weyl chamber a + . Since the description of these elements depends only on the root system ofP and not on the multiplicities, the number of poles of (P , o) coincides with the number of poles of (H, e), whereH is the connected simply connected compact simple Lie group whose root system is isomorphic to the one ofP .…”

“…In particular, Ad(P ) is not simply connected in this case. The description of the fundamental group of Ad(P ) due to Cartan [C] and Takeuchi [Tk] shows that the tangent Lie triple p of (P, o) must contain nonzero elements X with the property that ad(X) 3 = − ad(X) (see also [MQ1]). We call these elements extrinsically symmetric, because their isotropy orbits are extrinsically symmetric submanifolds in the Euclidean space p (see [F2], [EH]).…”

“…If X and Y are two different elements of a + of type I, then the corresponding centrioles K.(exp((1/2)πX).o) and K.(exp((1/2)πY ).o) are disjoint.Proof. In[MQ1, Lemma 4.4], the authors show that the poles exp(πX).o and exp(πY ).o of (P , o) are different. The claim follows from Corollary 6.…”

Natural Γ-symmetric structures on R-spaces

“…If I = {j} and ξ I = ξ j is extrinsic symmetric, then X I is a symmetric R-space in the sense of Takeuchi [T1] and Kobayashi and Nagano [KN] (see also [MQ,Lemma 2.1]).…”

Convexity of reflective submanifolds in symmetric $R$-spaces

“…For example symmetric R-spaces arise as certain spaces of shortest geodesics, namely as those centrioles (see [CN88]) that are formed by midpoints of shortest geodesics arcs joining a base point to a pole (see e.g. [MQ11a]). Reflective submanifolds in symmetric spaces include among others polars and centrioles (see e.g.…”