Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture

by Gilkey; Peter B.

• ISBN13: 9780849382772
• ISBN10: 0849382777
• Format: Hardcover
• Copyright: 1999-07-27
• Publisher: CRC Press

Summary

Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture explores cutting-edge theory and research on the form valued Laplacian in the class of smooth compact manifolds without boundary. The authors explore eigenform preservation in respect to pull back. They state that an eigenvalue changes only if the corresponding eigenvalue changes. This is a rare phenomenon, and the book offers a list of several never-resolved eigenvalue-change problems.

Table of Contents

Elliptic Operators
Introduction	1	(2)
The real and complex Laplace operators	3	(11)
Spinors	14	(11)
Spectral resolutions	25	(8)
Manifolds with boundary	33	(6)
Spectral invariants	39	(10)
The eta invariant	49	(9)
Computing the eta invariant	58	(13)
Differential Geometry
Introduction	71	(2)
Riemannian submersions	73	(9)
Characteristic classes	82	(5)
The geometry of sphere and principal bundles	87	(6)
The geometry of circle bundles	93	(7)
The Hopf fibration	100	(7)
The scalar curvature	107	(10)
Levi-Civita and spin connections	117	(4)
Positive Curvature
Introduction	121	(8)
Manifolds with positive Ricci curvature	129	(5)
Bordism and connective K theory	134	(14)
Calculations involving the eta invariant	148	(13)
The eta invariant and connective K theory	161	(14)
Computing connective K theory groups	175	(14)
Spectral Geometry of Riemannian Submersions
Introduction	189	(3)
Intertwining the coderivatives	192	(7)
The real Laplacian	199	(13)
The complex Laplacian	212	(12)
The spin Laplacian	224	(6)
Riemannian submersions with boundary	230	(8)
Heat trace and heat content	238	(4)
Unresolved questions	242	(1)
References
5.1 Introduction	243	(1)
5.2 Main bibliography	244	(17)
5.3 Bibliography of harmonic morphisms	261	(9)
5.4 Parabolic PDE bibliography	270	(3)
Notation	273	(4)
Index	277