Continuous families of isospectral metrics on simply connected manifolds

We construct continuous families of Riemannian metrics on certain simply connected manifolds with the property that the resulting Riemannian manifolds are pairwise isospectral for the Laplace operator acting on functions. These are the first examples of simply connected Riemannian manifolds without boundary which are isospectral, but not isometric. For example, we construct continuous isospectral families of metrics on the product of spheres S^4\times S^3\times S^3. The metrics considered are not locally homogeneous. For a big class of such families, the set of critical values of the scalar curvature function changes during the deformation. Moreover, the manifolds are in general not isospectral for the Laplace operator acting on 1-forms.Comment: 22 pages, published versio

Isospectral manifolds with different local geometries

We construct several new classes of isospectral manifolds with different local geometries. After reviewing a theorem by Carolyn Gordon on isospectral torus bundles and presenting certain useful specialized versions (Chapter 1) we apply these tools to construct the first examples of isospectral four-dimensional manifolds which are not locally isometric (Chapter 2). Moreover, we construct the first examples of isospectral left invariant metrics on compact Lie groups (Chapter 3). Thereby we also obtain the first continuous isospectral families of globally homogeneous manifolds and the first examples of isospectral manifolds which are simply connected and irreducible. Finally, we construct the first pairs of isospectral manifolds which are conformally equivalent and not locally isometric (Chapter 4).Comment: 59 pages, AMS-TeX. Habilitation thesis (2000), to appear in: J. Reine Angew. Mat

On the corner contributions to the heat coefficients of geodesic polygons

Let $\mathcal O$ be a compact Riemannian orbisurface. We compute formulas for the contribution of cone points of~$\mathcal O$ to the coefficient at $t^2$ of the asymptotic expansion of the heat trace of $\mathcal O$, the contributions at $t^0$ and $t^1$ being known from the literature. As an application, we compute the coefficient at $t^2$ of the contribution of interior angles of the form $\gamma=\pi/k$ in geodesic polygons in surfaces to the asymptotic expansion of the Dirichlet heat kernel of the polygon, under a certain symmetry assumption locally near the corresponding corner. The main novelty here is the determination of the way in which the Laplacian of the Gauss curvature at the corner point enters into the coefficient at $t^2$. We finish with a conjecture concerning the analogous contribution of an arbitrary angle $\gamma$ in a geodesic polygon.Comment: 21 page

Isospectral potentials and conformally equivalent isospectral metrics on spheres, balls and Lie groups

We construct pairs of conformally equivalent isospectral Riemannian metrics $\phi_1 g$ and $\phi_2 g$ on spheres $S^n$ and balls $B^{n+1}$ for certain dimensions $n$, the smallest of which is $n=7$, and on certain compact simple Lie groups. In the case of Lie groups, the metric $g$ is left-invariant. In the case of spheres and balls, the metric $g$ is not the standard metric but may be chosen arbitrarily close to the standard one. For the same manifolds $(M,g)$ we also show that the functions $\phi_1$ and $\phi_2$ are isospectral potentials for the Schr\"odinger operator $\hbar^2\Delta +\phi$. To our knowledge, these are the first examples of isospectral potentials and of isospectral conformally equivalent metrics on simply connected closed manifolds.Comment: 34 pages, AMS-TeX; revised subsection 5.

Local symmetry of harmonic spaces as determined by the spectra of small geodesic spheres

We show that in any harmonic space, the eigenvalue spectra of the Laplace operator on small geodesic spheres around a given point determine the norm $|\nabla R|$ of the covariant derivative of the Riemannian curvature tensor in that point. In particular, the spectra of small geodesic spheres in a harmonic space determine whether the space is locally symmetric. For the proof we use the first few heat invariants and consider certain coefficients in the radial power series expansions of the curvature invariants $|R|^2$ and $|Ric|^2$ of the geodesic spheres. Moreover, we obtain analogous results for geodesic balls with either Dirichlet or Neumann boundary conditions.Comment: 18 pages, LaTeX. Added a few lines in the introduction, corrected a few typos. Final version. Accepted for publication in GAF

Spectral isolation of bi-invariant metrics on compact Lie groups

We show that a bi-invariant metric on a compact connected Lie group $G$ is spectrally isolated within the class of left-invariant metrics. In fact, we prove that given a bi-invariant metric $g_0$ on $G$ there is a positive integer $N$ such that, within a neighborhood of $g_0$ in the class of left-invariant metrics of at most the same volume, $g_0$ is uniquely determined by the first $N$ distinct non-zero eigenvalues of its Laplacian (ignoring multiplicities). In the case where $G$ is simple, $N$ can be chosen to be two.Comment: 10 pages, new title, revised abstract and introduction, minor typos corrected, to appear in Ann. Inst. Fourier (Grenoble