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Spectrally distinguishing symmetric spaces II

Emilio A. Lauret, Juan Sebastián Rodríguez

Abstract

The action of the subgroup $\operatorname{G}_2$ of $\operatorname{SO}(7)$ (resp.\ $\operatorname{Spin}(7)$ of $\operatorname{SO}(8)$) on the Grassmannian space $M=\frac{\operatorname{SO}(7)}{\operatorname{SO}(5)\times\operatorname{SO}(2)}$ (resp.\ $M=\frac{\operatorname{SO}(8)}{\operatorname{SO}(5)\times\operatorname{SO}(3)}$) is still transitive. We prove that the spectrum (i.e.\ the collection of eigenvalues of its Laplace-Beltrami operator) of a symmetric metric $g_0$ on $M$ coincides with the spectrum of a $\operatorname{G}_2$-invariant (resp.\ $\operatorname{Spin}(7)$-invariant) metric $g$ on $M$ only if $g_0$ and $g$ are isometric. As a consequence, each non-flat compact irreducible symmetric space of non-group type is spectrally unique among the family of all currently known homogeneous metrics on its underlying differentiable manifold.

Spectrally distinguishing symmetric spaces II

Abstract

The action of the subgroup of (resp.\ of ) on the Grassmannian space (resp.\ ) is still transitive. We prove that the spectrum (i.e.\ the collection of eigenvalues of its Laplace-Beltrami operator) of a symmetric metric on coincides with the spectrum of a -invariant (resp.\ -invariant) metric on only if and are isometric. As a consequence, each non-flat compact irreducible symmetric space of non-group type is spectrally unique among the family of all currently known homogeneous metrics on its underlying differentiable manifold.

Paper Structure

This paper contains 16 sections, 9 theorems, 78 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The case Gr(2,7)
  4. Homogeneous metrics for $\operatorname{Gr}_{{2}}(\mathbb R^{{7}})$
  5. The tricky term for $\operatorname{Gr}_{{2}}(\mathbb R^{{7}})$
  6. Some low Laplace eigenvalues of $\operatorname{Gr}_{{2}}(\mathbb R^{{7}})$
  7. Spectral uniqueness for $\operatorname{Gr}_{{2}}(\mathbb R^{{7}})$
  8. Homogeneous Einstein metrics on $\operatorname{Gr}_{{2}}(\mathbb R^{{7}})$
  9. The case Gr(3,8)
  10. Root systems for $\operatorname{Gr}_{{3}}(\mathbb R^{{8}})$
  11. Homogeneous metrics for $\operatorname{Gr}_{{3}}(\mathbb R^{{8}})$
  12. Casimir scalars and the tricky term for $\operatorname{Gr}_{{3}}(\mathbb R^{{8}})$
  13. Presentations of the standard and the spin representation
  14. Some low Laplace eigenvalues of $\operatorname{Gr}_{{3}}(\mathbb R^{{8}})$
  15. Spectral uniqueness for $\operatorname{Gr}_{{3}}(\mathbb R^{{8}})$
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $G/H$ denote the second presentation of the symmetric space $M=\bar{G}/\bar{K}$ of any of the cases in eq:espaciossimetricos. If a $G$-invariant metric on $M$ is isospectral to a symmetric metric on $M$, then they are isometric.

Theorems & Definitions (24)

  • Theorem 1.2
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • ...and 14 more