Tetra and Didi, the cosmic spectral twins

Doyle, Peter G¹ ; Rossetti, Juan Pablo²

¹ Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755-3551, USA
² FaMAF – Ciem, Universidad Nacional de Córdoba, Argentina

Geometry & topology, Tome 8 (2004) no. 3, pp. 1227-1242.

Résumé

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.

Détail

DOI : 10.2140/gt.2004.8.1227

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

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