Classification Theory of Riemannian Manifolds Harmonic, quasiharmoni
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605 Leo Sario Mitsuru Nakai
Cecilia Wang Lung Ock Chung
Classification Theory of Riemannian Manifolds
Harmonic, quasiharmonic and biharmonic functions
Springer-Verlag Berlin Heidelberg NewYork 1977
Authors
Leo Sario Department of Mathematics University of California Los Angeles, CA 90024 USA
Cecilia Wang Department of Mathematics Arizona State University Tempe, AZ 85281 USA
Mitsuru Nakai Department of Mathematics Nagoya Institute of Technology Gokiso, Showa, Nagoya 466 Japan
Lung Ock Chung Department of Mathematics North Carolina State University Raleigh, NC 2?60? USA
Library of Congress Cataloging in Publication Data
Main e~try under title: Classification theory of Riemannian manifolds. (Lecture notes in mathematics ; 605) Bibliography: p. Includes indexes. 1. Harmonic functions. 2. Riemannian manifolds. I. Sario, Leo. II. Series: Lecture notes in mathematics (Berlin) ; 605. QA3.L28 no. 605 cQ~05~ 510t.Ss ~515'.533 77-22197
AMS Subject Classifications (1970): 31 BXX
tSBN 3-540-08358-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-38?-08358-8 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1977 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
To Angus E. Taylor
TABLE OF CONTENTS Preface and Historical Note CHAPTER 0 ~CE-BELTRAMI
~.
OPERATOR
Riemannian manifolds
12.
1. l,
Covariant and contravariant vectors
12
1.2.
Metric tensor
13
1.3.
Laplace-Beltrami operator
16
Harmonic forms
18
2.1.
Differential p-forms
18
2.2.
Hodge operator
2O
2.3.
Exterior derivative and coderivative
21
2.4.
Laplace-Beltrami operator
22
CHAPTER I HARMONIC FUNCTIONS §l.
Relations
ON = OGN < OB~ p 2
by
N-space
But in other problems, such as the strictness of
and
punctured
N < 0HD, N 0HB
the
higher dimensions brought in challenging difficulties that were only recently overcome.
The main gain in the shift to Riemannian manifolds was the availability
of new aspects that were not meaningful on abstract Riem~nn surfaces.
The
Lp
finiteness of the function and the completeness of the manifold are typical of these.
An account of this fourth phase of classification theory is given in
Chapter I of the present monograph. Fifth phase To anderstand the inauguration of the fifth phase of classification theory, the biharmonic classification of Riemanniau manifolds, we have to go back to the origin of biharmonic functions and to Airy, Astronomer Royal.
In fact, at this
point we intentionally go somewhat beyond the topic at hand, as we are her
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