Carolyn S. Gordon

One cannot hear the shape of a drum

We use an extension of Sunadas theorem to construct a nonisometric pair of isospectral simply connected domains in the Euclidean plane, thus answering negatively Kacs question, can one hear the shape of a drum? In order to construct simply connected examples, we exploit the observation that an orbifold whose underlying space is a simply connected manifold with boundary need not be simply connected as an orbifold

Chapter 6 – Survey of Isospectral Manifolds

Spectral geometry is the study of the relationship between the spectrum and geometry of M . This chapter focuses on the inverse spectral problem— the extent to which a spectrum encode the geometry. The chapter reviews the primary techniques for obtaining geometric information from the spectrum and describes some of the known spectral invariants. The chapter discusses spectral rigidity results and the structure of isospectral sets of metrics, in particular, the question of whether the collection of metrics with a given spectrum is compact in a natural topology. The chapter discusses the construction of isospectral, nonisometric metrics. Examples of such metrics provide the means for identifying specific geometric invariants that are not spectrally determined. The known methods for proving that two metrics are isospectral are: (1) explicit computation, (2) representation theoretic or trace formula methods, and (3) the use of Riemannian submersions with totally geodesic fibers. The correspondence between the classical dynamics of a Riemannian manifold, that is, the geodesic flow, and the quantum dynamics suggests a relationship between the Laplace isospectrality of manifolds and the symplectic conjugacy of their geodesic flows.

Homogeneous Riemannian manifolds whose geodesics are orbits

Let M be a homogeneous Riemannian manifold and G the isometry group of M. Then M can be viewed as a coset space G/H with a left-invariant Riemannian metric. M is said to be a g. o. manifold if every geodesic in M is an orbit of a one-parameter subgroup of G. In [KV1], O. Kowalski and L. Vanhecke showed that every g. o. manifold is a D’Atri space; i. e. the local geodesic symmetries are volume preserving up to sign. (See the survey article by Kowalski, Prufer and Vanhecke in this volume for a discussion of D’Atri spaces and additional comments about g. o. manifolds.) The simplest Riemannian homogeneous spaces are the naturally reductive manifolds; these form a subclass of the g. o. manifolds. The definition of naturally reductive (see §1) involves a purely algebraic condition on the isometry group.

Kähler structures on compact solvmanifolds

In a previous paper, the authors proved that the only compact nilmanifolds F\G which admit Kahler structures are tori. Here we consider a more general class of homogeneous spaces F\G, where G is a completely solvable Lie group and F is a cocompact discrete subgroup. Necessary conditions for the existence of a Kahler structure are given in terms of the structure of G and a homogeneous representative w of the Kahler class in H2(F\G; R). These conditions are not sufficient to imply the existence of a Kahler structure. On the other hand, we present examples of such solvmanifolds that have the same cohomology ring as a compact Kahler manifold. We do not know whether some of these solvmanifolds admit Kahler structures.

NEW HOMOGENEOUS EINSTEIN METRICS OF NEGATIVE RICCI CURVATURE

We construct new homogeneous Einstein spaces with negativeRicci curvature in two ways: First, we give a method for classifying andconstructing a class of rank one Einstein solvmanifolds whose derivedalgebras are two-step nilpotent. As an application, we describe anexplicit continuous family of ten-dimensional Einstein manifolds with atwo-dimensional parameter space, including a continuous subfamily ofmanifolds with negative sectional curvature. Secondly, we obtain newexamples of non-symmetric Einstein solvmanifolds by modifying thealgebraic structure of non-compact irreducible symmetric spaces of rankgreater than one, preserving the (constant) Ricci curvature.

Isospectral deformations of metrics on spheres

Abstract.We construct non-trivial continuous isospectral deformations of Riemannian metrics on the ball and on the sphere in Rn for every n≥9. The metrics on the sphere can be chosen arbitrarily close to the round metric; in particular, they can be chosen to be positively curved. The metrics on the ball are both Dirichlet and Neumann isospectral and can be chosen arbitrarily close to the flat metric.

Isospectral deformations of negatively curved Riemannian manifolds with boundary which are not locally isometric

To what extent does the eigenvalue spectrum of the Laplace-Beltrami operator on a compact Riemannian manifold determine the geometry of the manifold? We give examples of isospectral manifolds with different local geometry including continuous families of isospectral negatively curved manifolds with boundary as well as various pairs of manifolds. The latter illustrate that the spectrum does not determine whether a manifold with boundary has negative curvature, whether it has constant Ricci curvature, and whether it has parallel curvature tensor, and the spectrum does not determine whether a closed manifold has constant scalar curvature.

Isospectral families of conformally equivalent Riemannian metrics

Two Riemannian manifolds are said to be isospectral if their associated Laplace operators have the same spectrum. Many examples of pairs and of continuous families of isospectral manifolds have appeared in recent years; see [Ber] for a discussion. The first examples of pairs of conformally equivalent isospectral manifolds appeared in [BT], and a general construction was given by Brooks, Perry, and Yang [BPY]. The purpose of this note is to construct continuous families of isospectral conformally equivalent manifolds, using the method of [BPY]. We begin with an example: Let F be the Lie group of all 7 x 7 matrices of the form

The inaudible geometry of nilmanifolds.

SummaryWe show that isospectral deformations of compact Riemannian two-step nilmanifolds can be systematically detected by simple changes in the behavior of their geodesics, in spite of the fact that the length spectrum (which measures the lengths of all closed geodesics) remains constant.

Spectral Geometry on Nilmanifolds

Two Riemannian manifolds are said to be isospectral if the associated Laplace-Beltrami operators have the same spectrum. Riemannian nilmanifolds have provided a rich source of examples of isospectral manifolds, exhibiting a wide variety of different phenomena. In particular, there exist continuous families of isospectral, nonisometric nil-manifolds, isospectral nilmanifolds for which the Laplacians acting on one-forms are not isospectral, and isospectral nilmanifolds that are not even locally isometric. This article reviews three different methods for constructing isospectral nilmanifolds and examines the geometry of resulting examples.