Geometry & Topology

Tetra and Didi, the cosmic spectral twins

Peter G Doyle and Juan Pablo Rossetti

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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

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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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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