The Funk transform as a Penrose transform

Abstract

The Funk transform is the integral transform from the space of smooth even functions on the unit sphere S²[subset or is implied by][open face R]³ to itself defined by integration over great circles. One can regard this transform as a limit in a certain sense of the Penrose transform from [open face C][open face P]₂ to [open face C][open face P]*ast;₂. We exploit this viewpoint by developing a new proof of the bijectivity of the Funk transform which proceeds by considering the cohomology of a certain involutive (or formally integrable) structure on an intermediate space. This is the simplest example of what we hope will prove to be a general method of obtaining results in real integral geometry by means of complex holomorphic methods derived from the Penrose transform.