Topological Restrictions on Double Fibrations and Radon Transforms

Given two manifolds X and Y, the topological concept double fibration defines two integral Radon transforms R: Cf(X)^ C°°(Y) and R': C^r")-» C °°(X). For every x e X the double fibration specifies submanifolds of Y, Gx, all diffeomorphic to each other. For g E C °°( Y), x £ X, the transform R 'g(x) integrates g over Gx in a specified measure. Let k be the codimension of Gx in Y. Under the Bolker assumption, we show that k = 1, 2, 4, or 8. Furthermore if k = 1 then every Gx is diffeomorphic to S"~l or RP"~l, if it = 8 then Gx is homeomorphic to Ss. In the other cases Gx is a cohomology projective space. This shows that the manifolds Gx which occur are all similar to the Gx for the classical Radon transforms.