19 research outputs found
A manifold structure for the group of orbifold diffeomorphisms of a smooth orbifold
For a compact, smooth C^r orbifold (without boundary), we show that the
topological structure of the orbifold diffeomorphism group is a Banach manifold
for finite r \ge 1 and a Frechet manifold if r=infty. In each case, the local
model is the separable Banach (Frechet) space of C^r (C^infty, resp.)
orbisections of the tangent orbibundle.Comment: 26 pages, 2 figures, final versio
When is a trigonometric polynomial not a trigonometric polynomial?
As an application of B\'ezout's theorem from algebraic geometry, we show that
the standard notion of a trigonometric polynomial does not agree with a more
naive, but reasonable notion of trigonometric polynomial.Comment: 3 page
On the notions of suborbifold and orbifold embedding
The purpose of this article is to investigate the relationship between
suborbifolds and orbifold embeddings. In particular, we give natural
definitions of the notion of suborbifold and orbifold embedding and provide
many examples. Surprisingly, we show that there are (topologically embedded)
smooth suborbifolds which do not arise as the image of a smooth orbifold
embedding. We are also able to characterize those suborbifolds which can arise
as the images of orbifold embeddings. As an application, we show that a
length-minimizing curve (a geodesic segment) in a Riemannian orbifold can
always be realized as the image of an orbifold embedding.Comment: 11 pages. Final Version. arXiv admin note: text overlap with
arXiv:1205.115
Elementary orbifold differential topology
Taking an elementary and straightforward approach, we develop the concept of a regular value for a smooth map f:O→P between smooth orbifolds O and P. We show that Sardʼs theorem holds and that the inverse image of a regular value is a smooth full suborbifold of O. We also study some constraints that the existence of a smooth orbifold map imposes on local isotropy groups. As an application, we prove a Borsuk no retraction theorem for compact orbifolds with boundary and some obstructions to the existence of real-valued orbifold maps from local model orbifold charts
On the existence of infinitely many closed geodesics on orbifolds of revolution
Using the theory of geodesics on surfaces of revolution, we introduce the
period function. We use this as our main tool in showing that any
two-dimensional orbifold of revolution homeomorphic to S^2 must contain an
infinite number of geometrically distinct closed geodesics. Since any such
orbifold of revolution can be regarded as a topological two-sphere with metric
singularities, we will have extended Bangert's theorem on the existence of
infinitely many closed geodesics on any smooth Riemannian two-sphere. In
addition, we give an example of a two-sphere cone-manifold of revolution which
possesses a single closed geodesic, thus showing that Bangert's result does not
hold in the wider class of closed surfaces with cone manifold structures.Comment: 21 pages, 4 figures; for a PDF version see
http://www.calpoly.edu/~jborzell/Publications/publications.htm
The Stratified Structure of Spaces of Smooth Orbifold Mappings
We consider four notions of maps between smooth C^r orbifolds O, P with O
compact (without boundary). We show that one of these notions is natural and
necessary in order to uniquely define the notion of orbibundle pullback. For
the notion of complete orbifold map, we show that the corresponding set of C^r
maps between O and P with the C^r topology carries the structure of a smooth
C^\infty Banach (r finite)/Frechet (r=infty) manifold. For the notion of
complete reduced orbifold map, the corresponding set of C^r maps between O and
P with the C^r topology carries the structure of a smooth C^\infty Banach (r
finite)/Frechet (r=infty) orbifold. The remaining two notions carry a
stratified structure: The C^r orbifold maps between O and P is locally a
stratified space with strata modeled on smooth C^\infty Banach (r
finite)/Frechet (r=infty) manifolds while the set of C^r reduced orbifold maps
between O and P locally has the structure of a stratified space with strata
modeled on smooth C^\infty Banach (r finite)/Frechet (r=infty) orbifolds.
Furthermore, we give the explicit relationship between these notions of
orbifold map. Applying our results to the special case of orbifold
diffeomorphism groups, we show they inherit the structure of C^\infty Banach (r
finite)/Frechet (r=infty) manifolds. In fact, for r finite they are topological
groups, and for r=infty they are convenient Frechet Lie groups.Comment: 31 pages, 2 figures; corrected and expande
Orbifolds with Lower Ricci Curvature Bounds
We show that the first betti number of a compact Riemannian orbifold with Ricci curvature and diameter is bounded above by a constant , depending only on dimension, curvature and diameter. In the case when the orbifold has nonnegative Ricci curvature, we show that the is bounded above by the dimension , and that if, in addition, , then is a flat torus
Whose limit is it anyway?
In a tongue-in-cheek manner, we investigate the notion of limit. We illustrate some of its shortcomings and show that addressing these shortcomings can often lead to unexpected consequences