Spectral and geometric bounds on 2-orbifold diffeomorphism type (original) (raw)

Determination of the Topological Structure of an Orbifold by its Group of Orbifold Diffeomerphisms

Journal of Lie Theory, 2003

We show that the topological structure of a compact, locally smooth orbifold is determined by its orbifold dieomorphism group. Let Di r (O) denote the C r orbifold dieomorphisms of an orbifold O. Suppose that : Di r Orb (O1) ! Di r Orb (O2) is a group isomorphism between the the orbifold dieomorphism groups of two orbifolds O1 and O2 . We show that is induced by a homeomorphism h : XO1! XO2 , where XO denotes the under- lying topological space of O. That is, ( f) = hfh 1 for all f 2 Di r(O1). Furthermore, if r > 0, then h is a C r manifold dieomorphism when restricted to the complement of the singular set of each stratum.

Riemannian Geometry of Orbifolds

Also, 1 would like to thank Robert Greene and Shiu-Yuen Cheng for their valuable suggestions and for showing me what a beautiful subject geometry is. Thanks also go to the faculty and st.aff of the CCLA Department of Mathematics for making graduate study aL LTCLA a pleasure. I must thank those fellow graduate students and colleagues who haYe put up with me over the years.

A manifold structure for the group of orbifold diffeomorphisms of a smooth orbifold

Journal of Lie theory

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.

Orbifolds with lower Ricci curvature bounds

Proceedings of the American Mathematical Society

We show that the first betti number b(1)(O) = dim H-1(O,R) of a compact Riemannian orbifold O with Ricci curvature Ric(O) greater than or equal to - (n - 1)k and diameter diam(O) less than or equal to D is bounded above by a constant c(n, kD(2)) greater than or equal to 0, depending only on dimension, curvature and diameter. In the case when the orbifold has nonnegative Ricci curvature, we show that the b(1)(0) is bounded above by the dimension dim 0, and that if, in addition, b(1)(0)= dim 0, then O is a hat torus T-n.

Orbifold homeomorphism and diffeomorphism groups

Infinite Dimensional Lie Groups in …, 2002

Specifically, we define the notion of reduced and unreduced orbifold diffeomorphism groups. For the reduced orbifold diffeomorphism group we state and sketch the proof of the following recognition result: Let O 1 and O 2 be two compact, locally smooth orbifolds. Fix

The isometry groups of Riemannian orbifolds

Siberian Mathematical Journal, 2007

We prove that the isometry group I(N) of an arbitrary Riemannian orbifold N , endowed with the compact-open topology, is a Lie group acting smoothly and properly on N. Moreover, I(N) admits a unique smooth structure that makes it into a Lie group. We show in particular that the isometry group of each compact Riemannian orbifold with a negative definite Ricci tensor is finite, thus generalizing the well-known Bochner's theorem for Riemannian manifolds.

One cannot hear orbifold isotropy type

Archiv der Mathematik, 2006

We show that the isotropy types of the singularities of Riemannian orbifolds are not determined by the Laplace spectrum. Indeed, we construct arbitrarily large families of mutually isospectral orbifolds with different isotropy types. Finally, we show that the corresponding singular strata of two isospectral orbifolds may not be homeomorphic.

Topological Orbifolds

Proceedings of the American Mathematical Society, 1993

We show that two topologically homeomorphic orbifolds are also Lipshitz homeomorphic. We then prove that the L-class of a good orbifold with finite fundamental group depends only on the topological structure.