The geometry of drums

TL;DR

This paper introduces D-geometry, a new geometric framework that explains the construction of isospectral and length equivalent drums, revealing finite geometric structures underlying these spectral phenomena.

Contribution

The paper establishes a connection between D-geometry and the construction of isospectral drums, providing a new geometric perspective on spectral equivalence.

Findings

Any pair of length equivalent domains defines a D-geometry.

D-geometry underpins the Gassmann-Sunada method for constructing isospectral domains.

Finite geometrical phenomena control many examples of isospectral drums.

Abstract

We introduce the new concept of D-geometry (or "drum geometry"), which has been recently discovered by the author in \cite{KT-DRUMS} when constructing and classifying isospectral and length equivalent drums under certain constraints. We will show that any pair of length equivalent domains, and in particular any pair of isospectral domains (which makes one unable to "hear the shape of drums") which is constructed by the famous Gassmann-Sunada method, naturally defines a D-geometry, and that each D-geometry gives rise to such domains. One goal of this letter is to show that in the present theory of isospectral and length equivalent drums, many examples are controlled by finite geometrical phenomena in a very precise sense.