Geometric properties of natural operators defined by the Riemann curvature tensor / Peter B. Gilkey.

Format:

Book

Author/Creator:

Gilkey, Peter B.

Language:

English

Subjects (All):

Geometry, Riemannian.

Curvature.

Operator theory.

Physical Description:

1 online resource (316p. ) bibliog

Edition:

1st ed.

Place of Publication:

River Edge, NJ : World Scientific, c2001.

Language Note:

English

Summary:

This work presents results about the geometric consequences that follow if various natural operators defined in terms of the Riemann curvature tensor are assumed to have constant eigenvalues or constant Jordan normal form in the appropriate domains of definition.

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.

Contents:

Algebraic curvature tensors; the skew-symmetric curvature operator; the Jacobi operator; controlling the eigenvalue structure.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references (p. 293-302) and index.

ISBN:

9786611948016

9781281948014

1281948012

9789812799692

9812799699

OCLC:

815754176