Isospectral hyperbolic surfaces have matching geodesics

Details Isospectral hyperbolic surfaces have matching geodesics Isospectral hyperbolic surfaces have matching geodesics

2006-05-30

arxiv.org/abs/math/0605765v2

New York J. Math. 14 (2008) 193-2004

Mathematics

Differential Geometry

Version dated 29 April 2008; GNU FDL

Scientific paper

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.

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