Tetra and Didi, the cosmic spectral twins

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.

Freely redistributable version (GNU FDL). Differs only very slightly from the version published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper33.abs.html

References in corpus (2)

• Describing the platycosms
• Hearing the platycosms

Cited by in corpus (11)

• Hearing shapes of drums - mathematical and physical aspects of isospectrality
• Describing the platycosms
• Inflation, Large Branes, and the Shape of Space
• Isospectral hyperbolic surfaces have matching geodesics
• Non-strongly isospectral spherical space forms
• Prehomogeneous Affine Representations and Flat Pseudo-Riemannian Manifolds
• Geometric structures and the Laplace spectrum
• Geometric structures and the Laplace spectrum, part II
• Cubic Tessellations of the Helicosms
• The eta function and eta invariant of ${\mathbb{Z}}_{2^r}$-manifolds
• Isospectral orbifolds with different maximal isotropy orders