Geometric Properties of Natural Operators Defined by the Riemann Curvature Tensor - Hardcover

Peter B Gilkey

Hardcover

ISBN 10: 9810247524 ISBN 13: 9789810247522

Publisher: World Scientific Pub Co Inc, 2001

Synopsis

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.

About this title

Review

This clearly written monograph will be very useful for researchers and graduate students in this field. -- Mathematical Reviews, 2002

Edition details

• Publisher: World Scientific Pub Co Inc
• Publication date: 2001
• Language: English
• ISBN 10: 9810247524
• ISBN 13: 9789810247522
• Binding: Hardcover
• Edition number: 1
• Number of pages: 316