Geometry & Topology
- Geom. Topol.
- Volume 8, Number 3 (2004), 1227-1242.
Tetra and Didi, the cosmic spectral twins
We introduce a pair of isospectral but non-isometric compact flat 3–manifolds called (a tetracosm) and (a didicosm). The closed geodesics of and are very different. Where has two quarter-twisting geodesics of the shortest length, 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.
Geom. Topol., Volume 8, Number 3 (2004), 1227-1242.
Received: 17 July 2003
Revised: 20 September 2004
Accepted: 21 September 2004
First available in Project Euclid: 21 December 2017
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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