Minimum and Conjugate Points in Symmetric Spaces

Abstract

The purpose of this paper is to discuss conjugate points in symmetric spaces. Although the results are neither surprising nor altogether unknown, the author does not know of their explicit occurrence in the literature.Briefly, conjugate points in the tangent bundle to the tangent space at a point of a symmetric space are characterized in terms of the algebraic structure of the symmetric space. It is then shown that in the simply connected case the first conjugate locus coincides with the minimum (cut) locus. The interest in this last fact lies in its identification of a more or less locally and analytically defined set with one which includes all the topological interest of the space.