The Funk Transform

Let X be a Riemannian manifold of dimension n with the metric tensor g and Y = {Y} be a family of closed subvarieties of X of dimension k, 0 < k < n. For a continuous function f in X that decreases sufficiently fast at infinity, we define the integrals 3.1$$Mf\left( Y \right) = \int {_Y} fdV\left( Y \right),Y \in Y$$ where dV(Y) is the volume element on Y induced by the metric g. We call the function Mf (Y an integral transform of f. For a Euclidean space X and the family of hyperplanes we call this operator a Radon transform. We follow this terminology in any situation where the geometry is symmetric with respect to a transitive commutative group; another example: tori.