We construct isospectral pairs of Riemannian metrics on S 5 and on B 6 , thus lowering by three the minimal dimension of spheres and balls on which such metrics have been constructed previously (S n≥8 and B n≥9 ). We also construct continuous families of isospectral Riemannian metrics on S 7 and on B 8 . In each of these examples, the metrics can be chosen equal to the standard metric outside subsets of arbitrarily small volume.

Abstract. 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).

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 × T m , where T m is a torus of dimension m ≥ 2 and S n is a sphere of dimension n ≥ 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.

Abstract. The method of torus actions developed by the first and third authors yields examples of isospectral, non-isometric metrics on compact manifolds and isophasal, non-isometric metrics on non-compact manifolds. In contrast to most examples constructed by the Sunada method, the resulting examples have different local geometry. In this review we discuss insights into the inverse spectral problem gained through both of these approaches.

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 × S 3 × 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.