Geometric Properties of Natural Operators Defined by the Riemann Curvature Tensor

by Gilkey, Peter B.

• ISBN13: 9789810247522
• ISBN10: 9810247524
• Format: Hardcover
• Copyright: 2001-11-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

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 conse-quences 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 whos 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.

Table of Contents

Preface	v
Algebraic Curvature Tensors	1	(92)
Introduction	1	(3)
Results from linear algebra	4	(11)
Self-adjoint maps of a spacelike vector space	15	(4)
Clifford algebras and matrices	19	(3)
Natural operators	22	(7)
Algebraic curvature tensors	29	(5)
Einstein and k-stein algebraic curvature tensors	34	(5)
Properties of the curvature tensors R&phis;	39	(10)
Invariants of the orthogonal group	49	(5)
Natural operators with constant eigenvalues	54	(11)
The exponential map and Jacobi vector fields	65	(5)
Geometric realizations of algebraic curvature tensors	70	(5)
Schur problems	75	(3)
Space forms	78	(4)
Complex and para-complex space forms	82	(11)
The Skew-Symmetric Curvature Operator	93	(84)
Introduction	93	(5)
Examples	98	(4)
Rank 2 algebraic curvature tensors	102	(17)
Geometric realizations of rank 2 tensors	119	(4)
IP algebraic curvature tensors	123	(6)
Examples of IP manifolds	129	(5)
Classification of IP manifolds	134	(8)
Four dimensional geometry	142	(3)
Seven dimensional geometry	145	(3)
Eight dimensional geometry	148	(20)
Almost complex IP tensors	168	(7)
Higher order IP tensors	175	(2)
The Jacobi Operator	177	(62)
Introduction	177	(8)
Examples of Osserman tensors	185	(8)
Examples of higher order Osserman tensors	193	(8)
Rakic duality	201	(2)
The Osserman conjecture	203	(5)
Space forms and (para-) complex space forms	208	(15)
The higher order Jacobi operator	223	(10)
The Szabo operator	233	(6)
Controlling the Eigenvalue Structure	239	(54)
Introduction	239	(5)
Fiber bundles	244	(12)
Characteristic classes and K-theory	256	(9)
Symmetric vector bundles	265	(8)
Odd maps of constant rank	273	(20)
Bibliography	293	(10)
Index	303