Inverse spectral problems on a closed manifold

Abstract

In this paper we consider two inverse problems on a closed connected Riemannian manifold $(M,g)$. The first one is a direct analog of the Gel'fand inverse boundary spectral problem. To formulate it, assume that $M$ is divided by a hypersurface $Σ$ into two components and we know the eigenvalues $λ_j$ of the Laplace operator on $(M,g)$ and also the Cauchy data, on $Σ$, of the corresponding eigenfunctions $ϕ_j$, i.e. $ϕ_j|_Σ,\partial_νϕ_j|_Σ$, where $ν$ is the normal to $Σ$. We prove that these data determine $(M,g)$ uniquely, i.e. up to an isometry. In the second problem we are given much less data, namely, $λ_j$ and $ϕ_j|_Σ$ only. However, if $Σ$ consists of at least two components, $Σ_1, Σ_2$, we are still able to determine $(M,g)$ assuming some conditions on $M$ and $Σ$. These conditions are formulated in terms of the spectra of the manifolds with boundary obtained by cutting $M$ along $Σ_i$, $i=1,2$, and are of a generic nature. We consider also some other inverse problems on $M$ related to the above with data which is easier to obtain from measurements than the spectral data described.