.
The relationship between synthetic
geometry and convexity in Riemannian
manifolds
Doctor James Dibble
Western Illinois University
Abstract: A manifold is a topological space that’s locally
modeled on Euclidean space, in the sense that each point is
contained in an open set that’s homeomorphic to a ball in
Rn . A Riemannian manifold is one in which it’s possible to
sensibly talk about the lengths of tangent vectors, as well as
the angles between them, and these quantities vary smoothly.
By John Nash’s famous embedding theorem, Riemannian
manifolds are exactly higher-dimensional generalizations of
surfaces embedded in R3 .
Two basic quantities associated to a compact Riemannian
manifold M are its injectivity radius and its convexity radius.
The injectivity radius inj(M ) is the smallest positive number
such that no two distance-realizing curves of length less
than inj(M ) intersect exactly twice. The convexity radius
r(M ) is the smallest positive number such that any open
metric ball of radius at most r(M ) is convex in a strong sense.
An elementary result is that r(M ) ≤ 21 inj(M ), but there are
no examples in the literature where this inequality is strict.
r(M )
It will be shown in this talk that the ratio inj(M
may be made
)
arbitrarily small within the class of compact Riemannian manifolds of any fixed dimension at least two. The proof uses the
conditions of having no focal points or no conjugate points,
which are reminiscent of classical synthetic geometry, and is
inspired by a new characterization of the convexity radius.
DEPARTMENT
of
MATHEMATICS
Thursday,
October 8, 2015
3:45 p.m.
Morgan Hall 204
Refreshments
will be served at 3:30
p.m.