Convexity of reflective submanifolds in symmetric $R$-spaces

Abstract

We show that every reflective submanifold of a symmetric R-space is (geodesically) convex.Introduction.The main result in this article is the following.THEOREM 1. Reflective submanifolds of symmetric R-spaces are (geodesically) convex.We organized this article as follows.In Section 1, we define all notions used in Theorem 1. Reflective submanifolds in symmetric R-spaces are described in Section 2. The proof of Theorem 1 can be found in Section 3. In Section 4, we explain why the assumption "symmetric R-space" in Theorem 1 can not be generalized to all compact symmetric spaces.Symmetric R-spaces, introduced by Takeuchi and Nagano in the 1960s, form a class of compact symmetric spaces that have very peculiar geometric properties and appear in various contexts.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.[MQ12]).Reflective submanifolds in symmetric spaces include among others polars and centrioles (see e.g.[CN88,Na88,Qu11]).An iterative construction involving such centrioles has been used by Bott [Bo59] in the first proof of his famous periodicity result for the homotopy groups of classical Lie groups (see also [Mi69, § 23, 24] and [Mi88, § 7]).For the construction described in [MQ11, Sect.1.2], it is important that the distance between a base point and a pole in a centriole of certain R-spaces measured in the centriole is the same as the distance measured in the ambient R-space.This follows directly from Theorem 1.Theorem 1 also provides a conceptional proof of [NS91, Remark 3.2b] in the case where the ambient space is a symmetric R-space.1. Preliminaries.We first define the terminology used in Theorem 1. Reflective submanifolds.A reflective submanifold M of a Riemannian manifold P is a connected component of the fixed point set of an involutive isometry τ of P , that is τ 2 equals 2000