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Non-compact inaudibility of weak symmetry and commutativity via generalized Heisenberg groups

Teresa Arias-Marco, José Manuel Fernández-Barroso

Abstract

Two Riemannian manifolds are said to be isospectral if there exists a unitary operator which intertwines their Laplace-Beltrami operator. In this paper, we prove in the non-compact setting the inaudibility of the weak symmetry property and the commutative property using an isospectral pair of 23 dimensional generalized Heisenberg groups.

Non-compact inaudibility of weak symmetry and commutativity via generalized Heisenberg groups

Abstract

Two Riemannian manifolds are said to be isospectral if there exists a unitary operator which intertwines their Laplace-Beltrami operator. In this paper, we prove in the non-compact setting the inaudibility of the weak symmetry property and the commutative property using an isospectral pair of 23 dimensional generalized Heisenberg groups.
Paper Structure (3 sections, 8 theorems, 30 equations)

This paper contains 3 sections, 8 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Isospectrality on generalized Heisenberg groups
  3. Main results and conclusions

Key Result

Theorem 2.1

$N^{p,q}$ and $N^{p',q'}$ are isospectral if $p+q=p'+q'$. Moreover, they are locally isometric if and only if $(p',q')=\{(p,q),(q,p)\}$.

Theorems & Definitions (11)

  • Theorem 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Theorem 3.1: Ricci, R.85
  • Theorem 3.2: Berndt, Ricci, Vanhecke, BRV.98
  • Theorem 3.3: Riehm, Rie.84
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • ...and 1 more