Positively curved 7-dimensional manifolds

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...

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.

An Infinite Family of Distinct 7-manifolds Admitting Positively Curved Riemannian Structures

On One Family of 13-dimensional Closed Riemannian Positively Curved Manifolds

In the present paper we describe one family of closed Riemannian manifolds with positive sectional curvature. Now the list of known examples is not large (for instance, all known manifolds with dimension > 24 are diffeomorphic to compact rank one symmetric spaces) (we restrict ourselves only by pointing out simply connected manifolds) : 1) Berger described all normally homogeneous closed positi...