Differential Geometry and Mathematical Physics Lectures given at the
This volume contains the text of the lectures which were given at the Differential Geometry Meeting held at Liege in 1980 and at the Differential Geometry Meeting held at Leuven in 1981. The first of these meetings was more orientated toward mathematical
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MATHEMATICAL PHYSICS STUDIES A SUPPLEMENTARY SERIES TO LETTERS IN MATHEMATICAL PHYSICS
Editors: M. FLA TO, Universitt! de Dijon, France M. GUENIN, Institut de Physique Theorique, Geneva, Switzerland R. R+CZKA, Institute of Nuclear Research, Warsaw, Poland J. SIMON, UniversitedeDijon, France S. ULAM, University of Colorado, U.S.A.
Assistant Editor: J. C. CORTET, Universitede Dijon, France
Editorial Board: W. AMREIN, Institut de Physique Theorique, Geneva, Switzerland H. ARAKI, Kyoto University, Japan A. CONNES,l.H.E.S., France 1. FADDEEV, Steklov Institute ofMathematics, Leningrad, U.S.S.R. J. FROHLlCH,l.H.E.S., France C. FRONSDAL, UCLA, Los Angeles, U.S.A. I. M. GELFAND,Moscow State University, U.S.S.R. A. JAFFE, Harvard University, U.S.A. M. KAC, The Rockefeller University, New York, U.S.A. A. A. KIRILLOV, Moscow State University, U.S.S.R. A. DCHNEROWICZ, College de France, France E. H. DEB,Princeton University, U.S.A.
B. NAGEL,K.T.H., Stockholm, Sweden J. NIED ERLE, Institute ofPhysics CSAV, Prague, Czechoslovakia A. SALAM, International Center for Theoretical Physics, Trieste, Italy I. E. SEGAL,M.l.T., U.S.A. W. SCHM ID, Harvard University, U.S.A. D. STERNHEIMER, College de France, France I. T. TODOROV ,Institute of Nuclear Research, Sofia, Bulgaria VOLUME 3
Differential Geometry and Mathematical Physics Lectures given at the Meetings of the Belgian Contact Group on Differential Geometry held at Liege, May 2-3,1980 and at Leuven, February 6-8,1981 Edited by
M. CAHEN Un;versite Libre de Bruxelles, Belgium
M. DE WILDE Universite de Liege, Belgium
L. LEMAIRE Unfversite Libre de Bruxelles. Belgium
and
L. VANHECKE Katholieke Universiteit Leuven, Belgium
D. Reidel Publishing Company Dordrecht : Holland / Boston: U.S.A. / London: England
library of Congress Cataloging in Publication Data Main entry under title: Differential geometry and mathematical physics. (Mathematical physics studies; v. 3) Includes index. 1. Geometry, Differential-Addresses, essays, lectures. 2. Mathematical physics-Addresses, essays, lectures. 1. Cahen, M. (Michel), 1935. II. Belgian Contact Group on Differential
Geometry. III. Series. QC20.7.D52D538 1982 ISBN-13: 978-90-277-1508-1 DOl: 10.1007/978-94-009-7022-9
516.3'6 82-20535 1983 by D. Reidel PubliBhing Company.
27
Maxwell's fields. Explicitly: A = {~ : M + M diffeo. such that if F is a Coo2-form on M with dF
= of = 0
then d ~*F
= 0 ~*F = oJ.
This definition led us to consider the commutation relations
of the operators 0 and £X where X is a COO vector field on M and lx denotes the Lie derivative. One has
on M Buch that o£xw = £xow for all exact p-forms on M (where p is any fixed integer such that 1 ~ P ~ dim M). Then X is an infinitesimal isometry. 1) Let X be a vector
fie~d
2) A vector field X on an orientable manifold of even dimension m satisfies O£xw = 0 for any m/2 -form such that ow = 0 if and only if X is a conforma~ vector fie~d. To determine the invariance group of Maxwell's equation, we
use a) a theorem of Palais concerning finite dimensional Lie algebra
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