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The Funk transform as a Penrose transform

Published online by Cambridge University Press:  01 January 1999

TOBY N. BAILEY ,
MICHAEL G. EASTWOOD ,
A. ROD GOVER  and
LIONEL J. MASON
TOBY N. BAILEY
Affiliation:
Department of Mathematics and Statistics, University of Edinburgh, Edinburgh; e-mail: tnb@maths.ed.ac.uk
MICHAEL G. EASTWOOD
Affiliation:
Department of Pure Mathematics, University of Adelaide, Adelaide, South Australia; e-mail: meastwoo@maths.adelaide.edu.au
A. ROD GOVER
Affiliation:
School of Mathematics, Queensland University of Technology, Brisbane, Queensland, Australia; e-mail: r.gover@fsc.qut.edu.au
LIONEL J. MASON
Affiliation:
St Peter's College, Oxford; e-mail: lmason@maths.ox.ac.uk
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Abstract

The Funk transform is the integral transform from the space of smooth even functions on the unit sphere S2⊂ℝ3 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 [Copf ]ℙ2 to [Copf ]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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 125 , Issue 1 , January 1999 , pp. 67 - 81
Copyright
Cambridge Philosophical Society 1999

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Footnotes

This research was supported by the Australian Research Council and the Isaac Newton Institute for Mathematical Sciences. The last named author was supported by NATO collaborative research grant 950300.