The Blaschke conjecture and great circle fibrations of spheres

We construct an explicit diffeomorphism taking any fibration of a sphere by great circles into the Hopf fibration, using elementary geometry--indeed the diffeomorphism is a local (differential) invariant, algebraic in derivatives.Comment: 61 pages, 8 figures, corrected errors in the published versio

Lifting locally homogeneous geometric structures

We prove that under some purely algebraic conditions every locally homogeneous structure modelled on some homogeneous space is induced by a locally homogeneous structure modelled on a different homogeneous space.Comment: 9 page

Extension Phenomena for Holomorphic Geometric Structures

The most commonly encountered types of complex analytic G-structures and Cartan geometries cannot have singularities of complex codimension 2 or more.Comment: published versio

Summary of progress on the Blaschke conjecture

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 summarize the intuition behind this problem, the proof that such manifolds have the cohomology of compact rank one symmetric spaces, and the proof of the conjecture for homology spheres and homology real projective spaces. We also summarize what is known on the diffeomorphism, homeomorphism and homotopy types of such manifolds.Comment: 21 page