E-Book Overview
A central problem in differential geometry is to relate algebraic properties of the Riemann curvature tensor to the underlying geometry of the manifold. The full curvature tensor is in general quite difficult to deal with. This book presents results about the geometric consequences that follow if various natural operators defined in terms of the Riemann curvature tensor (the Jacobi operator, the skew-symmetric curvature operator, the Szabo operator, and higher order generalizations) are assumed to have constant eigenvalues or constant Jordan normal form in the appropriate domains of definition.
The book presents algebraic preliminaries and various Schur type problems; deals with the skew-symmetric curvature operator in the real and complex settings and provides the classification of algebraic curvature tensors whose skew-symmetric curvature has constant rank 2 and constant eigenvalues; discusses the Jacobi operator and a higher order generalization and gives a unified treatment of the Osserman conjecture and related questions; and establishes the results from algebraic topology that are necessary for controlling the eigenvalue structures. An extensive bibliography is provided. Results are described in the Riemannian, Lorentzian, and higher signature settings, and many families of examples are displayed.
E-Book Content
Geometric Properties of Natural Operators Curvature Tensor Peter B. Gilkey
World Scienti
Geometric Properties of Natural Operators Defined bq the Riemonn Curvature Tensor
Geometric Properties of Natural Operators Defined bq [be Riemann Curvature Tensor Peter B. Gilkey University of Oregon, USA
10 World Scientific ll
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GEOMETRIC PROPERTIES OF NATURAL OPERATORS DEFINED BY THE RIEMANN CURVATURE TENSOR Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN 981-02-4752-4
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Preface
A central problem in differential geometry is to relate algebraic properties of the Riemann curvature tensor to the underlying geometry of the manifold. The full curvature tensor is in general quite difficult to deal with. We will use the curvature tensor to define several natural associated operators; the Jacobi operator, the Szabo operator, and the skew-symmetric curvature operator are all natural operators of differential geometry which are defined in terms of the curvature tensor and its covariant derivative. We also consider other related operators. We shall discuss the geometric conditions which are imposed when we assume that one of these operators has constant eigenvalues. Chapter 1 of this book is devoted to algebraic preliminaries. Chapter 2 deals with the skew-symmetric curvature operator. Chapter 3 deals with the Jacobi and Szabo operators. Chapter 4 discusses results from algebraic topology which are needed previously. The first section in Chapters 1-4 contains a lengthy int