1,723 research outputs found
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
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