Isospectral hyperbolic surfaces have matching geodesics
Differential Geometry
2009-04-08 v2 Spectral Theory
Abstract
We show that if two closed hyperbolic surfaces (not necessarily orientable or even connected) have the same Laplace spectrum, then for every length they have the same number of orientation-preserving geodesics and the same number of orientation-reversing geodesics. Restricted to orientable surfaces, this result reduces to Huber's theorem of 1959. Appropriately generalized, it extends to hyperbolic 2-orbifolds (possibly disconnected). We give examples showing that it fails for disconnected flat 2-orbifolds.
Cite
@article{arxiv.math/0605765,
title = {Isospectral hyperbolic surfaces have matching geodesics},
author = {Peter G. Doyle and Juan Pablo Rossetti},
journal= {arXiv preprint arXiv:math/0605765},
year = {2009}
}
Comments
Version dated 29 April 2008; GNU FDL