Riemannian Isometry Groups Containing Transitive Reductive Subgroups.
The inaudible geometry of nilmanifolds.
Quantum Equivalent Magnetic Fields that Are Not Classically Equivalent
We construct pairs of compact Kähler-Einstein manifolds of complex dimension with the following properties: The canonical line bundle has Chern class , and for each positive integer the tensor powers and are isospectral for the bundle Laplacian associated with the canonical connection, while and – and hence and – are not homeomorphic. In the context of geometric quantization, we interpret these examples as magnetic fields which are quantum equivalent but not classically equivalent....
Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn't reveal
Let be a -dimensional compact Riemannian manifold. We show that the spectrum of the Hodge Laplacian acting on -forms does not determine whether the manifold has boundary, nor does it determine the lengths of the closed geodesics. Among the many examples are a projective space and a hemisphere that have the same Hodge spectrum on 1- forms, and hyperbolic surfaces, mutually isospectral on 1-forms, with different injectivity radii. The Hodge -spectrum also does not distinguish orbifolds from manifolds....
Symplectic rigidity of geodesic flows on two-step nilmanifolds
Inverse spectral results on even dimensional tori
Given a Hermitian line bundle over a flat torus , a connection on , and a function on , one associates a Schrödinger operator acting on sections of ; its spectrum is denoted . Motivated by work of V. Guillemin in dimension two, we consider line bundles over tori of arbitrary even dimension with “translation invariant” connections , and we address the extent to which the spectrum determines the potential . With a genericity condition, we show that if the connection is invariant under...
Spectral isolation of bi-invariant metrics on compact Lie groups
We show that a bi-invariant metric on a compact connected Lie group is spectrally isolated within the class of left-invariant metrics. In fact, we prove that given a bi-invariant metric on there is a positive integer such that, within a neighborhood of in the class of left-invariant metrics of at most the same volume, is uniquely determined by the first distinct non-zero eigenvalues of its Laplacian (ignoring multiplicities). In the case where is simple, can be chosen to be two....
Isospectral deformations of closed riemannian manifolds with different scalar curvature
We construct the first examples of continuous families of isospectral Riemannian metrics that are not locally isometric on closed manifolds , more precisely, on , where is a torus of dimension and is a sphere of dimension . 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.
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