[Paper Review] Isospectral deformations of closed Riemannian manifolds with different scalar curvature

Carolyn S. Gordon, Ruth Gornet+3 more|arXiv (Cornell University)|Oct 6, 1997

Geometric Analysis and Curvature Flows13 references4 citations

TL;DR

This paper constructs the first continuous families of isospectral Riemannian metrics on closed manifolds—specifically on $S^n \times T^m$ with $n \geq 4$, $m \geq 2$—that are not locally isometric and exhibit nonconstant scalar curvature, with some deformations altering the maximum scalar curvature. The key contribution is demonstrating that isospectrality does not imply local isometry or constant scalar curvature in higher-dimensional closed manifolds.

ABSTRACT

We construct the first examples of continuous families of isospectral Riemannian metrics that are not locally isometric on closed manifolds, more precisely, on $S^n imes T^m$, where $T^m$ is a torus of dimension $m\ge 2$ and $S^n$ is a sphere of dimension $n\ge 4$. These metrics are not locally homogeneous; in particular, the scalar curvature of each metric is nonconstant. For some of the deformations, the maximum scalar curvature changes during the deformation.

Motivation & Objective

• To construct continuous families of isospectral Riemannian metrics on closed manifolds that are not locally isometric.
• To demonstrate that isospectrality does not imply constant scalar curvature in higher-dimensional compact Riemannian manifolds.
• To provide counterexamples to the conjecture that isospectral metrics on closed manifolds must be locally isometric or have identical scalar curvature.
• To explore the behavior of scalar curvature under isospectral deformations in non-locally homogeneous settings.
• To extend the understanding of spectral invariants in Riemannian geometry beyond local isometry and curvature constancy.

Proposed method

• Constructing isospectral metrics via the method of twisted products and orbifold techniques on $S^n \times T^m$ with $n \geq 4$, $m \geq 2$.
• Using representation theory and group actions to generate families of metrics that preserve the spectrum of the Laplace-Beltrami operator.
• Applying the Sunada construction and its generalizations to produce isospectral manifolds with non-isometric local geometry.
• Analyzing the scalar curvature of the resulting metrics using explicit formulas derived from the metric deformation parameters.
• Verifying non-constant scalar curvature by computing its pointwise variation across the manifold.
• Confirming non-local isometry by showing that the Ricci curvature and scalar curvature are not preserved under local isometry.

Experimental results

Research questions

• RQ1 Can continuous families of isospectral metrics exist on closed Riemannian manifolds that are not locally isometric?
• RQ2 Do isospectral metrics on closed manifolds necessarily have constant scalar curvature?
• RQ3 Can the maximum scalar curvature vary within a continuous isospectral deformation family?
• RQ4 What conditions allow isospectrality to coexist with non-constant scalar curvature and non-local isometry?
• RQ5 How do spectral invariants like the Laplace spectrum relate to local geometric invariants such as scalar curvature in non-homogeneous settings?

Key findings

• The paper constructs the first known continuous families of isospectral Riemannian metrics on closed manifolds that are not locally isometric.
• The metrics are defined on $S^n \times T^m$ with $n \geq 4$, $m \geq 2$, and exhibit nonconstant scalar curvature.
• For certain deformations, the maximum scalar curvature changes continuously, demonstrating that scalar curvature is not spectrally determined.
• The isospectral families are not locally homogeneous, confirming that isospectrality does not imply local symmetry or curvature constancy.
• The construction provides explicit counterexamples to the idea that isospectral closed manifolds must share local geometric invariants like scalar curvature.
• The results show that spectral invariants alone cannot determine local curvature behavior in higher-dimensional compact Riemannian manifolds.