We prove a new comparison lemma for Jacobi fields that exploits Wilking's transverse
Jacobi equation. In contrast to standard Riccati and Jacobi comparison theorems, there are …

A connected Riemannian manifold M has constant vector curvature ε, denoted by cvc (ε), if
every tangent vector v∈ TM lies in a 2-plane with sectional curvature ε. When the sectional …

We discuss the rigidity (or lack thereof) imposed by different notions of having an abundance
of zero curvature planes on a complete Riemannian $3 $-manifold. We prove a rank rigidity …

This paper presents hyperbolic rank rigidity results for rank 1, nonpositively curved spaces.
Let $ M $ be a compact, rank 1 manifold with nonpositive sectional curvature and suppose …

The Blaschke conjecture claims that every compact Riemannian manifold whose injectivity
radius equals its diameter is, up to constant rescaling, a compact rank one symmetric space …

We consider rigidity properties of compact symmetric spaces $ X $ with metric $ g_0 $ of
rank one. Suppose $ g $ is another Riemannian metric on $ X $ with sectional curvature …

In this paper we extend the concept of a conjugate point in a Riemannian manifold to
geodesic spaces. In particular, we introduce symmetric conjugate points and ultimate …

A construction is described that associates to each positive smooth function F: S 1→ ℝ a
smooth Riemannian metric g F on SL 2 (ℝ)≅ ℝ 2× S 1 that is complete and curvature …

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 …

In a 1983 paper with Frank Warner, we proved that the space of all great circle fibrations of
the 3-sphere S^ 3 deformation retracts to the subspace of Hopf fibrations, and so has the …