On Infinitesimal Inverse Spectral Geometry

Abstract

Spectral geometry is the field of mathematics which concerns relationships between geometric structures of manifolds and the spectra of canonical differential operators. Inverse Spectral Geometry in particular concerns the geometric information that can be recovered from the knowledge of such spectra.

A deep link between inverse spectral geometry and sampling theory has recently been proposed. Specifically, it has been shown that the very shape of a Riemannian manifold can be discretely sampled and then reconstructed up to a cutoff scale. In the context of Quantum Gravity, this means that, in the presence of a physically motivated ultraviolet cuttoff, spacetime could be regarded as simultaneously continuous and discrete, in the sense that information can.

In this thesis, we look into the properties of the Laplace-Beltrami operator on a compact Riemannian manifold with no boundary. We discuss the behaviour of its spectrum regarding a perturbation of the Riemannian structure. Specifically, we concern ourselves with infinitesimal inverse spectral geometry, the inverse spectral problem of locally determining the shape of a Riemannian manifold. We discuss the recenSpectral geometry is the field of mathematics which concerns relationships between geometric structures of manifolds and the spectra of canonical differential operators. Inverse Spectral Geometry in particular concerns the geometric information that can be recovered from the knowledge of such spectra.

A deep link between inverse spectral geometry and sampling theory has recently been proposed. Specifically, it has been shown that the very shape of a Riemannian manifold can be discretely sampled and then reconstructed up to a cutoff scale. In the context of Quantum Gravity, this means that, in the presence of a physically motivated ultraviolet cuttoff, spacetime could be regarded as simultaneously continuous and discrete, in the sense that information can.

In this thesis, we look into the properties of the Laplace-Beltrami operator on a compact Riemannian manifold with no boundary. We discuss the behaviour of its spectrum regarding a perturbation of the Riemannian structure. Specifically, we concern ourselves with infinitesimal inverse spectral geometry, the inverse spectral problem of locally determining the shape of a Riemannian manifold. We discuss the recently presented idea that, in the presence of a cutoff, a perturbation of a Riemannian manifold could be uniquely determined by the knowledge of the spectra of natural differential operators. We apply this idea to the specific problem of determining perturbations of the two dimensional flat torus through the knowledge of the spectrum of the Laplace-Beltrami operator.tly presented idea that, in the presence of a cutoff, a perturbation of a Riemannian manifold could be uniquely determined by the knowledge of the spectra of natural differential operators. We apply this idea to the specific problem of determining perturbations of the two dimensional flat torus through the knowledge of the spectrum of the Laplace-Beltrami operator.