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Topological restrictions on double fibrations and Radon transforms


Author: Eric Todd Quinto
Journal: Proc. Amer. Math. Soc. 81 (1981), 570-574
MSC: Primary 58G15; Secondary 32M15, 44A05, 55R25
DOI: https://doi.org/10.1090/S0002-9939-1981-0601732-0
MathSciNet review: 601732
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0601732-0
Keywords: Generalized Radon transform, double fibration, Hopf invariant, projective space
Article copyright: © Copyright 1981 American Mathematical Society

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