A Survey of Inverse Spectral Results

The existence of the Laplace-Beltrami operator has allowed mathematicians to carry out Fourier analysis on Riemannian manifolds [2]. We recall that the Laplace-Beltrami operator ∆ on a compact Riemannian manifold has a discrete set of eigenvalues {λj} ∞ j=1, which satisfies λj → ∞ as j → ∞. This is known as the spectrum of the Laplace-Beltrami operator. Inverse spectral geometry studies how much of the geometry of the manifold is determined by this spectrum. The purpose of this paper is to survey some of the results in this area and to give indications of some of the techniques used to prove them. As is customary in the literature, we will use the term the spectrum of the manifold with the same meaning as the term the spectrum of the Laplace-Beltrami operator on the manifold. We will also use the following terminology: a spectral invariant is a quantity which is determined by the spectrum of the manifold, and two Riemannian manifolds are called isospectral, if their spectra, counting multiplicities, coincide.