Some questions on integral geometry on Riemmanian manifolds

Abstract

It is a surprising but known fact that an L¹ function on Rⁿ is determined once all its spherical averages with centres coming from certain "smal" sets are known. We generalise this result to complete connected real-analytic Riemannian manifolds with a real-analytic metric (with some curvature restrictions), where superisothermal sets play the same role as balls do in the case of Euclidean space.