Courtesy Southern Evening Echo, Southampton, U.K. Evan Tom Davies 1904-1973 Topics in Differential Geometry Edited by HANNO RUND WILLIAM F. FORBES DEPARTMENT OF. MATHEMATICS FACULTY OF MATHEMATICS UNIVERSITY OF ARIZONA UNIVERSITY OF WATERLOO TUCSON, ARIZONA WATERLOO, ONTARIO, CANADA and DEPARTMENT OF APPLIED MATHEMATICS UNIVERSITY OF WATERLOO WATERLOO, ONTARIO, CANADA In memory of Evan Tom Davies ACADEMIC PRESS New York San Francisco London 1976 A Subsidiary of Harcourt Brace Jovanovich, Publishers Topics in Differential Geometry Edited by HANNO RUND WILLIAM F. FORBES DEPARTMENT OF. MATHEMATICS FACULTY OF MATHEMATICS UNIVERSITY OF ARIZONA UNIVERSITY OF WATERLOO TUCSON, ARIZONA WATERLOO, ONTARIO, CANADA and DEPARTMENT OF APPLIED MATHEMATICS UNIVERSITY OF WATERLOO WATERLOO, ONTARIO, CANADA In memory of Evan Tom Davies ACADEMIC PRESS New York San Francisco London 1976 A Subsidiary of Harcourt Brace Jovanovich, Publishers COPYRIGHT © 1976, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. Ill Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NW1 Library of Congress Cataloging in Publication Data Main entry under title: Topics in differential geometry. Includes bibliographies and index. CONTENTS : Clark, R. S. Obituary, Evan Tom Davies. - Bompiani, E. Reminiscences of E. T. Davies.-Anderson. I. M. The uniqueness of the neutrino energy-momentum tensor and the Einstein-Weyl equations. [etc.] 1. Geometry, Differential-Addresses, essays, lectures. 2. Davies, Evan Tom, (date) . I. Davies, Evan Tom, (date) II. Rund, Hanno. HI. Forbes, William F. QA641.T6 516'.36 75-44114 ISBN 0-12-602850-8 PRINTED IN THE UNITED STATES OF AMERICA List of Contributors Numbers in parentheses indicate the pages on which the authors' contributions begin. M. S. AL-BORNEY (29), Mathematics Department, Faculty of Science, Damascus University, Damascus, Syria IAN M. ANDERSON (15), Department of Mathematics, University of Arizona, Tucson, Arizona E. BOMPIANI* (9), Rome, Italy F. BRICKELL (29), Department of Mathematics, The University, South- ampton, United Kingdom R. S. CLARK (1, 29), Department of Mathematics, The University, Southampton, United Kingdom GREGORY WALTER HORNDESKI (45), Department of Applied Mathe- matics, University of Waterloo, Waterloo, Ontario, Canada ANDRÉ LICHNEROWICZ (57), College de France, Paris, France DAVID LOVELOCK (87), Department of Mathematics, University of Arizona, Tucson, Arizona, and Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada M. A. McKIERNAN (99), Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada KAM-PING MOK (157), Department of Mathematics, University of Hong Kong, Hong Kong HANNO RUND (111), Department of Mathematics, University of Arizona, Tucson, Arizona, and Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada A. G. WALKER (135), Thorncroft, Thornton Common Road, Wirral, United Kingdom T. J. WILLMORE (149), Department of Mathematics, Science Laboratories, University of Durham, United Kingdom ♦Deceased ix X LIST OF CONTRIBUTORS YUNG-CHOW WONG (157), Department of Mathematics, University of Hong Kong, Hong Kong KENTARO YANO (173), Department of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo, Japan Preface Professor Evan Tom Davies died unexpectedly on October 8, 1973 in Waterloo, Ontario, Canada. He had devoted the last years of his life to the Faculty of Mathematics of the University of Waterloo, and consequently, a few days after his death, several of his friends at Waterloo expressed the desire to pay tribute to the memory of this man who had won their deep respect and admiration. Since the thrust of Professor Davies' research activities had been directed primarily at differential geometry, it was agreed to publish a collection of articles in this field, written by some of his former friends and associates. It was thus that the present volume came into existence; we hope only that it represents, at least in some small measure, a worthy memorial to our late friend. Dr. R. S. Clark of the University of Southampton kindly consented to let us reproduce his detailed obituary notice of Professor Davies, which appeared in 1974 in the Bulletin of the London Mathematical Society, and we acknowledge with gratitude the Society's permission to reprint this article. Our thanks are due also to Professor A. G. Walker, F.R.S., of the University of Liverpool, and Professor David Lovelock o fthe University of Arizona, for their generous assistance with various editorial tasks. Finally, it is a pleasure to acknowledge our indebtedness to the publishers for their unfailing courtesy and cooperation which they extended to us at all times. XI 1 Evan Tom Davies R. S. CLARK Evan Tom Davies was born on 24 September 1904 at the small village of Pencader, Carmarthenshire, the younger son of a farmer, Thomas Davies, and his wife Elizabeth. His parents were Welsh-speaking and throughout his life Ianto, as he was affectionately known to his many friends, retained a passionate regard for the Welsh culture. He received his secondary schooling at Llandyssul County School. From here he went on to University College, Aberystwyth and graduated with honors in applied mathematics in 1924. He then moved to University College, Swansea where he had the distinction of being the first student to graduate with honors in pure mathematics. He stayed on to write a thesis on n-dimensional geometry, and for this he was awarded an M.Sc. in 1926. Acting on the advice of Paul Dienes, who was lecturing in Swansea at this time, Ianto then turned his attention to the absolute differential calculus. This was being developed as one of the chief mathematical tools in the theory of relativity. So in August 1926 he traveled to Rome to study under Tullio Levi-Civita. This was the beginning of many eventful visits to Europe, and in later years he had many tales to tell of his experiences. He used to recall that he was arrested three times, although he was lucky enough to escape gaol. On the first occasion in Rome, a party of students was making merry in the Colosseum and since many nationalities were represented among 1 Reprinted from Bull. London Math. Soc. 6 (1974), 370-376, with permission. 1 2 R. S. CLARK them, they decided that each should sing a song of his native land. As Wales begins with a W, lanto had the honor of singing last. The strains of "Land of My Fathers" were ringing strongly through the Colosseum when the police arrived. Nearly all the examinations in Rome were oral and open to the public. By now lanto spoke fluent Italian. He was examined by eleven professors on a day when the temperature was over 100°F. However, he achieved success and was awarded his doctorate. But overwork had brought on a temporary breakdown in health, and another year was to pass before he was completely fit and ready to travel again. In 1928 he left home again to spend a year in Paris at the Sorbonne and the Collège de France. As in Italy, he met many young differential geometers who later became leading mathematicians and were to remain life-long friends. He was appointed assistant lecturer at King's College, London in 1930 and so commenced his career as a University teacher and creative mathematician. In those days young lecturers had to be prepared to teach the junior classes. lanto became very accomplished in teaching generally, and especially in dealing with the Engineers who could become very lively at times. He traveled extensively in Europe during the vacations and became fluent in five languages. Every Christmas he would join a party of friends for winter sports, and he soon acquired some ability as a skier. In those pre-war days in Switzerland the atmosphere in the evening was elegant and relaxed. lanto was an excellent raconteur, and these were the hours in which he shone most brilliantly. He also became a keen Wimbledon tennis fan, and for many years he would take an early place in the queue for the Centre Court on the First Thursday and the Second Wednesday of the tournament. All this time his mathematical research was developing. He was promoted to lecturer in 1935 and became a University Reader in 1946. A few months later lanto accepted the Chair of Mathematics at University College, Southampton. At that time he was the only professor of mathematics in the country who had not studied at either Oxford or Cambridge. With his natural gaiety, his flair for dancing and for college functions generally, he quickly became a highly popular and respected figure. In those early days his department was small and its annual meeting would take place somewhere in the country after a strawberry and cream tea provided by the professor. In 1952 University College became Southampton University, and the department grew from a staff of six to a multi- professorial one of over sixty. He played an increasing role in the higher administration of the University and acted as Deputy Vice-Chancellor (1954-1957) and Dean of Science (1965-1967). EVAN TOM DAVIES 3 During the early years of the War, King's College had been evacuated to Bristol. Ianto made friends as readily here as elsewhere, and in 1941 he married Margaret Helen Picton. It was a devastating blow to him when she died in London in 1944. Happily, however, after moving to Southampton he met Hilda Gladys Boyens. They married in 1955 and had one son, Geraint. Their home at once became a center for Ianto's hospitality, and the department still enjoyed its strawberry and cream teas, though its official meetings were held elsewhere. He enjoyed many associations outside the University, and perhaps the most important to him was the Rotary move- ment, of which he was a very loyal supporter. In his travels he met Rotarians and addressed Rotary Clubs all over the world. He took a keen and friendly interest in all his students and research flourished under his guidance. In 1954 and 1962 he made extensive tours in South East Asia visiting many countries, meeting many mathematicians, most of them already personal friends, and addressing many Universities. What he saw and heard about the economic difficulties which face the developing countries concerned him deeply, and he was often to speak about this on his return. These tours had their lighter side as well. On one occasion he found himself billed to appear on the Japanese television to discuss geometrical problems relating to the design of the kimono with his old friend and colleague Kentaro Yano. On his retirement in 1969 at the age o f65 the University recognized its indebtedness to him by conferring on him the title of Emeritus Professor, and establishing the E. T. Davies Prize for the best honors graduate in mathematics. But the retirement was in name only, for he then proceeded to Canada and became a professor at the University o fCalgary. He quickly established himself in the Department of Mathematics, and enjoyed both the countryside and the way of life in that hospitable department. Finally in 1971 he accepted a similar position in the University of Waterloo, Ontario. Once more he soon made many friends and was prominent in their Friday afternoon ritual of saying good-by to the week over a mug of beer in the Faculty Club. It was on this campus that he was taken ill on 8 October, 1973, and within hours he died. He was then in his sixty-ninth year. Thus he completed his life, as he would have wished, in harness. Several of Ianto's friends have kindly helped me prepare this notice. For help with the following appreciation of his mathematical work I am particularly grateful to Dr. F. Brickell. Suppose that J is a geometric object field on a differentiable manifold M. A local transformation φ: U -+M of M induces on U a local geometric object field φ^ J of the same type as J itself, φ is said to be a local auto- morphism of J if J = φ+ J on U. A vector field X on M is called an infinitesimal transformation of M. Each point m of M admits a neighborhood