Geometry & Topology

Tetra and Didi, the cosmic spectral twins

Peter G Doyle and Juan Pablo Rossetti

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Abstract

We introduce a pair of isospectral but non-isometric compact flat 3–manifolds called Tetra (a tetracosm) and Didi (a didicosm). The closed geodesics of Tetra and Didi are very different. Where Tetra has two quarter-twisting geodesics of the shortest length, Didi has four half-twisting geodesics. Nevertheless, these spaces are isospectral. This isospectrality can be proven directly by matching eigenfunctions having the same eigenvalue. However, the real interest of this pair – and what led us to discover it – is the way isospectrality emerges from the Selberg trace formula, as the result of a delicate interplay between the lengths and twists of closed geodesics.

Article information

Source
Geom. Topol., Volume 8, Number 3 (2004), 1227-1242.

Dates
Received: 17 July 2003
Revised: 20 September 2004
Accepted: 21 September 2004
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883466

Digital Object Identifier
doi:10.2140/gt.2004.8.1227

Mathematical Reviews number (MathSciNet)
MR2087082

Zentralblatt MATH identifier
1091.58021

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds 58J53: Isospectrality
Secondary: 11F72: Spectral theory; Selberg trace formula 53C22: Geodesics [See also 58E10]

Keywords
flat structure 3–manifold platycosm Laplace spectrum isospectral Selberg trace formula closed geodesic

Citation

Doyle, Peter G; Rossetti, Juan Pablo. Tetra and Didi, the cosmic spectral twins. Geom. Topol. 8 (2004), no. 3, 1227--1242. doi:10.2140/gt.2004.8.1227. https://projecteuclid.org/euclid.gt/1513883466


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