Topological restrictions on double fibrations and Radon transforms

Abstract:

Given two manifolds $X$ and $Y$, the topological concept double fibration defines two integral Radon transforms $R:C_0^\infty (X) \to {C^\infty }(Y)$ and ${R^t}:{C^\infty }(Y) \to {C^\infty }(X)$. For every $x \in X$ the double fibration specifies submanifolds of $Y$, ${G_x}$, all diffeomorphic to each other. For $g \in {C^\infty }(Y)$, $x \in X$, the transform ${R^t}g(x)$ integrates $g$ over ${G_x}$ in a specified measure. Let $k$ be the codimension of ${G_x}$ in $Y$. Under the Bolker assumption, we show that $k = 1,2,4$, or 8. Furthermore if $k = 1$ then every ${G_x}$ is diffeomorphic to ${S^{n - 1}}$ or $R{P^{n - 1}}$, if $k = 8$ then ${G_x}$ is homeomorphic to ${S^8}$. In the other cases ${G_x}$ is a cohomology projective space. This shows that the manifolds ${G_x}$ which occur are all similar to the ${G_x}$ for the classical Radon transforms.