Geometric structures and the Laplace spectrum, part II

TL;DR

This paper investigates how the Laplace spectrum encodes local geometry of three-manifolds, revealing constraints on isospectral pairs and classifying locally homogeneous metrics, with implications for physical chemistry.

Contribution

It provides a classification of isospectral locally homogeneous three-manifolds, describes the isometry group of compact Lie groups with left-invariant metrics, and shows spectral distinguishability of certain metrics.

Findings

Isospectral non-isometric elliptic three-manifolds are rare and have specific properties.

Any collection of isospectral locally homogeneous metrics on an elliptic three-manifold has at most two isometry classes.

Left-invariant metrics on SO(3) and S^3 are spectrally distinguishable.

Abstract

We continue our exploration of the extent to which the spectrum encodes the local geometry of a locally homogeneous three-manifold and find that if $(M,g)$ and $(N,h)$ are a pair of locally homogeneous, locally non-isometric isospectral three-manifolds, where $M$ is an elliptic three-manifold, then $(1)$ $N$ is also an elliptic three-manifold, $(2)$ $M$ and $N$ have fundamental groups of different orders, $(3)$ $(M,g)$ and $(N,h)$ both have non-degenerate Ricci tensors and $(4)$ the metrics $g$ and $h$ are sufficiently far from a metric of constant sectional curvature. We are unaware of any such isospectral pair and such a pair could not arise via the classical Sunada method. As part of the proof, we provide an explicit description of the isometry group of a compact simple Lie group equipped with a left-invariant metric---improving upon the results of Ochiai-Takahashi and Onishchik---which we use to classify the locally homogeneous metrics on an elliptic three-manifold $\Gamma \backslash S^3$ and we determine that any collection of isospectral locally homogeneous metrics on an elliptic three-manifold consists of at most two isometry classes that are necessarily locally isometric. In particular, the left-invariant metrics on $\operatorname{SO}(3)$ (respectively, $S^3$) can be mutually distinguished via their spectra. The previous statement has the following interpretation in terms of physical chemistry: the moments of inertia of a molecule can be recovered from its rotational spectrum.