Contemporary MATHEMATICS 63 Integral Geometry

CoNTEMPORARY MATHEMATICS 63

Integral Geometry

AMS-IMS-SIAM Summer Research Conference on Integral Geometry August 12-18, 1984 Bowdoin College, Brunswick, Maine

Robert L. Bryant Sigurdur Helgason R. 0. Wells, Jr. Editors http://dx.doi.org/10.1090/conm/063 Recent Titles in This Series

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Integral Geometry

AMS-IMS-SIAM Summer Research Conference on Integral Geometry August 12-18, 1984 Bowdoin College, Brunswick, Maine

Robert L. Bryant Victor Guillemin Sigurdur Helgason R. 0. Wells, Jr. Editors

American Mathematical Society Providence. Rhode Island Editorial Board Irwin Kra, W. H. Jaco managing editor Gerald J. Janusz Thomas F. Banchoff Jan Mycielski M. Salah Baouendi Johannes C. C. Nitsche Alan D. Weinstein

The AMS-MAA-SIAM Joint Summer Research Conference in the Mathematical Sciences on Integral Geometry was held at Bowdoin College, Brunswick, Maine on August 12-18, 1984 with support from the National Science Foundation, Grant DMS-8218075. 1991 Mathematics Subject Classification. Primary 53C65; Secondary 53C35, 52A22, 60D05, 43A85.

Library of Congress Cataloging-in-Publication Data AMS-IMS-SIAM Joint Summer Research Conference in the Mathematical Sciences on Integral Geometry {1984: Bowdoin College) Integral Geometry. (Contemporary mathematics, ISSN 0271-4132; v. 63) Bibliography: p. 1. Geometry, Integral-Congresses. I. Bryant, Robert L. II. American Mathematical Soci- ety. III. Institute of Mathematical Statistics. IV. Society for Industrial and Applied Mathe- matics. V. Title. VI. Series: Contemporary mathematics (American Mathematical Society); v. 63. QA649.A49 1984 516.3'62 86-28902 ISBN 0-8218-5071-7 (alk. paper)

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© Copyright 1987 by the American Mathematical Society. All rights reserved. All rights reserved except those granted to the United States Government. Printed in the United States of America. This volume was printed from author prepared copy. § The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. 0 Printed on recycled paper. 10 9 8 7 6 54 3 2 00 99 98 97 96 95 TABLE OF CONTENTS

Preface vii

Spectral synthesis on symmetric spaces By Carlos A. Berenstein 1 The finite Radon transform By Ethan D. Balker 27 Hyperfunctions in representation theory and mathematical

phys~cs By Edward G. Dunne 51 The exponential Radon transform By David V. Finch and Alexander Hertle 67 Integral geometry as geometry and as analysis By S. G. Gindikin 75 Euclidean Radon transforms: ranges and restrictions By Eric L. Grinberg 109 Perspectives in integral geometry By Victor Guillemin 135 Some results on Radon transforms, Huygen's principle and x-ray transforms

By Sigurdur Helgason 151 Classical integral geometry in Riemannian homogeneous spaces

By Ralph Howard 179 Differential operators and Cartan motion groups

By Kenneth D. Johnson 205

v vi TABLE OF CONTENTS

An L2-cohomology analogue of the Penrose transform for the oscillator representation By Lisa A. Mantini 221 Injectivity of rotation invariant Radon transforms on complex hyperplanes in en By Eric Todd Quinto 245 On overdetermined systems associated with integral geometry transforms in the real projective space By Radu RO§U 261 The Pompeiu problem in exterior domains in symmetric spaces By Mehrdad Shahshahani and Alladi Sitaram 267 Curvature integrals and Chern classes of singular varieties By Theodore Shifrin 279 Hypothesis testing in integral geometry: guessing the shape of a plane domain By Peter Waksman 299 Non-linear integral transforms By Richard S. Ward 307 Integral geometry and twistor theory By R. 0. Wells, Jr. 317 Some inverse problems of potential theory By Lawrence Zalcman 337 PREFACE

In the summer of 1984 a one-week conference on Integral Geom- etry was held at Bowdoin College in Brunswick, Maine. It was one of a series of conferences organized under the auspices of the AMS-SIAM-IMS Summer Research Conference Series. The organizing committee consisted of Robert Bryant (chairman), Victor Guillemin, Sigurdur Helgason, and R. 0. Wells, Jr. The papers in this volume are contributions by participants whose papers, after refereeing, were considered suitable for inclusion. They range from purely expository to quite technical papers, and as a whole represent a good survey of contemporary work in this area. The topic of Integral Geometry as such is not nearly so uni- versally known as its counterpart, Differential Geometry, and most of the major results in this area are part of the twentieth century, while Differential Geometry has its very strong nineteenth century tradition going back to Gauss and Riemann. The conference brought together researchers from three different areas of contemporary research in topics involving integral geometry. The first area deals with new developments concerning· the classical problems of computing geometric invariants by statistical averaging procedures (to oversimplify greatly). The second area relates to a circle of ideas concerning what is now called the Radon transform, going back to the seminal work of Funk and Radon around 1916-1917. The third area deals with integral- viii PREFACE

geometric transforms which are now being used in the study of field equations in mathematical physics, work which derives from the twistor geometry of Penrose. Some of these overlapping areas involve group-representation theoretic problems also. The papers in this volume represent contributions from all of these areas. It is the hope of the organizers of this conference (who are simultaneously the editors of this volume) that bringing together these three branches of integral geometry will be a stimulant for further developments in these diverse areas. The questions asked in each case are often quite different, but the basic integral ·geometric idea lies at the basis of all of them. Fundamentally, integral geometry involves a transformation of information from one space to a derived space, using an integration procedure (or statistical averaging process) . The most famous example is the well-known CAT-scan now used routinely in medical diagnosis, and this is, of course, related to the original Radon transform involving integration of functions over lines in 3-space. Understanding the range, and the inversion of such integral-geometric transforms, is a major technical problem in the area, as a number of the papers below give evidence for. In the modern era, differential geometry is intertwined with the global theory of manifolds, although its initial impetus was to local problems. Hopefully, the developments in integral geometry in its many guises can give equally deep insight into the structure of both the global and local nature of manifolds (and their generalizations), and the functions (and their generalizations) which reside on these spaces. We would like to express our gratitude to Russell Poley for typing these papers. He passed away before the volume was com- PREFACE ix pleted, and we want to express our warmest sympathy and thanks to his widow, Anita Poley, who, under very trying circumstances, helped us complete the volume.

Robert L. Bryant Houston, Texas Victor Guillemin Cambridge, Hassachusetts Sigurdur Helgason Cambridge, Massachusetts R. 0. Wells, Jr. Houston, Texas 1 October 1986 Recent Titles in This Series (Continued from the front of this publication)

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(See the AMS catalog for earlier titles) ISBN 0-8218-5071 -7