On the topology of positively curved Kaehler manifolds, I

Positively Curved Combinatorial 3-Manifolds

We present two theorems in the “discrete differential geometry” of positively curved spaces. The first is a combinatorial analog of the Bonnet-Myers theorem: • A combinatorial 3-manifold whose edges have degree at most five has edgediameter at most five. When all edges have unit length, this degree bound is equivalent to an angle-deficit along each edge. It is for this reason we call such space...

Lecture on Kaehler manifolds

Positively Curved Manifolds with Large Spherical Rank

Rigidity results are obtained for Riemannian d-manifolds with sec > 1 and spherical rank at least d − 2 > 0. Conjecturally, all such manifolds are locally isometric to a round sphere or complex projective space with the (symmetric) Fubini– Study metric. This conjecture is verified in all odd dimensions, for metrics on dspheres when d 6= 6, for Riemannian manifolds satisfying the Rakić duality p...

Positively Curved Manifolds with Large Conjugate Radius

Let M denote a complete simply connected Riemannian manifold with all sectional curvatures ≥ 1. The purpose of this paper is to prove that when M has conjugate radius at least π/2, its injectivity radius and conjugate radius coincide. Metric characterizations of compact rank one symmetric spaces are given as applications.

Existence of closed geodesics on positively curved Finsler manifolds

For non-reversible Finsler metrics of positive flag curvature on spheres and projective spaces we present results about the number and the length of closed geodesics and about their stability properties. 2000 MSC classification: 53C22; 53C60; 58E10