On the Diameter of Compact Homogeneous Riemannian Manifolds

Abstract

Let M be a compact Riemannian manifold. The diameter d(M) of M is defined to be the maximum of d(p, q) p, q e M, where d( , ) denotes the distance function on M induced by the Riemannian metric. The main purpose of this paper is to find a positive constant d such that the diameter d(M)^d when the sectional curvature K^l. In this paper we consider the case that the manifold M is homogeneous. In [3] the author proved that d = n/2 if the manifold has a big isotropy subgroup. It has been left to study the case that the isotropy subgroup is finite. Hence we shall mainly study invariant metrics on a Lie group and prove that the number rf>0.23 if the sectional curvature K=£Q (Theorem 5.1).