The Funk transform as a Penrose transform

The Punk transform is the integral transform from the space of smooth even functions on the unit sphere S-2 subset of R-3 to itself defined by integra tion over great circles. One can regard this transform as a limit in a cert ain sense of the Penrose transform from CP2 to CP2* . We exploit this viewp oint by developing a new proof of the bijectivity of the Funk transform whi ch proceeds by considering the cohomology of a certain involutive (or forma lly integrable) structure on an intermediate space. This is the simplest ex ample of what we hope will prove to be a general method of obtaining result s in real integral geometry by means of complex holomorphic methods derived from the Penrose transform.