Great sphere foliations and manifolds with curvature bounded above

Abstract

The survey is devoted to Toponogov's conjecture, that {\it if a complete simply connected Riemannian manifold with sectional curvature $\le 4$ and injectivity radius $\ge \pi /2$ has extremal diameter $\pi /2$, then it is isometric to CROSS}. In Section 1 the relations of problem with geodesic foliations of a round sphere are considered, but the proof of conjecture on this way is not complete. In Section 2 the proof based on recent results and methods for topology and volume of Blaschke manifolds is given.

arXiv commentAMS-TeX v 2.1, 13 pages

References in corpus (1)

• The Radius Rigidity Theorem for Manifolds of Positive Curvature

Cited by in corpus (1)

• Spherical rank rigidity and Blaschke manifolds