FRACTIONAL CALDER<acute accent>ON PROBLEM ON A CLOSED RIEMANNIAN MANIFOLD

Given a fixed alpha is an element of (0, 1), we study the inverse problem of recovering the isometry class of a smooth closed and connected Riemannian manifold (M, g), given the knowledge of a source -to -solution map for the fractional Laplace equation (-Delta g)alpha u = f on the manifold subject to an arbitrarily small observation region O where sources can be placed and solutions can be measured. This can be viewed as a non -local analogue of the well known anisotropic Calder ' on problem that is concerned with the limiting case alpha = 1. In this paper, we solve the non -local problem under the assumption that the a priori known observation region O belongs to some Gevrey class while making no geometric assumptions on the inaccessible region of the manifold, namely M \ O. Our proof is based on discovering a connection to a variant of Carlson's theorem in complex analysis that reduces the inverse problem to Gel'fand inverse spectral problem.