A N Application of the Morse Theory to the Topology of Lie Groups

2. The Freudenthal theorem for symmetric spaces Our primary interest is in the classical compact groups; nevertheless, it is essential for our method to consider the larger family of compact symmetric spaces. A compact homogeneous Riemannian manifold M is called symmetric if it admits an 'inverse operation', i.e. if M admits an isometry, keeping a point P e M fixed, and whose differential at P is — 1. These geometric generalizations of the compact groups seem to be the class of spaces to which the Morse theory is most applicable. The reason is that, on such a space, conjugate points have global implications: