On the Topology of Riemannian Manifolds Where the Conjugate Points Have a Similar Distribution as in Symmetric Spaces of Rank 1 by Wilhelm Klingenberg

1. Manifolds similar to spheres. 1.1. Let S=S be the w-dimensional sphere, endowed with the usual metric of constant Riemannian curvature 1. Let G=(p(s)), 0^s< oo, be a geodesic ray in S, s being the arc length. Then the conjugate points of £(0) on G occur at S = IT, I a positive integer, with multiplicity n — 1. Let G be a geodesic ray in a Riemannian manifold M=M of dimension n. The following condition may be interpreted, at least for k = n — 1, as saying that the first k conjugate points on G are similarly distributed as on the sphere S: (2, k) There are no conjugate points in the interval [0, w[and at least k conjugate points in [7r, 2TT[, each counted by its multiplicity, 1.2. The following proposition gives a sufficient, but not necessary condition for the validity of (2, n —1). For the proof see Morse [5].