1,601 research outputs found
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
Nonlinear Gravitons, Null Geodesics, and Holomorphic Disks
We develop a global twistor correspondence for pseudo-Riemannian conformal
structures of signature (++--) with self-dual Weyl curvature. Near the
conformal class of the standard indefinite product metric on S^2 x S^2, there
is an infinite-dimensional moduli space of such conformal structures, and each
of these has the surprising global property that its null geodesics are all
periodic. Each such conformal structure arises from a family of holomorphic
disks in CP_3 with boundary on some totally real embedding of RP^3 into CP_3.
An interesting sub-class of these conformal structures are represented by
scalar-flat indefinite K\"ahler metrics, and our methods give particularly
sharp results in this more restrictive setting.Comment: 56 pages, LaTeX2
A direct approach to quaternionic manifolds
The recent definition of slice regular function of several quaternionic
variables suggests a new notion of quaternionic manifold. We give the
definition of quaternionic regular manifold, as a space locally modeled on
, in a slice regular sense. We exhibit some significant classes
of examples, including manifolds which carry a quaternionic affine structure.Comment: 13 page
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