Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture: 1st Edition (Hardback) book cover

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

1st Edition

By Peter B. Gilkey, John V Leahy, JeongHyeong Park

CRC Press

296 pages

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Hardback: 9780849382772
pub: 1999-07-27
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Description

This cutting-edge, standard-setting text explores the spectral geometry of Riemannian submersions. Working for the most part with the form valued Laplacian in the class of smooth compact manifolds without boundary, the authors study the relationship-if any-between the spectrum of Dp on Y and Dp on Z, given that Dp is the p form valued Laplacian and pi: Z ® Y is a Riemannian submersion.

After providing the necessary background, including basic differential geometry and a discussion of Laplace type operators, the authors address rigidity theorems. They establish conditions that ensure that the pull back of every eigenform on Y is an eigenform on Z so the eigenvalues do not change, then show that if a single eigensection is preserved, the eigenvalues do not change for the scalar or Bochner Laplacians. For the form valued Laplacian, they show that if an eigenform is preserved, then the corresponding eigenvalue can only increase. They generalize these results to the complex setting as well. However, the spinor setting is quite different. For a manifold with non-trivial boundary and imposed Neumann boundary conditions, the result is surprising-the eigenvalues can change.

Although this is a relatively rare phenomenon, the authors give examples-a circle bundle or, more generally, a principal bundle with structure group G where the first cohomology group H1(G;R) is non trivial. They show similar results in the complex setting, show that eigenvalues can decrease in the spinor setting, and offer a list of unsolved problems in this area.

Moving to some related topics involving questions of positive curvature, for the first time in mathematical literature the authors establish a link between the spectral geometry of Riemannian submersions and the Gromov-Lawson conjecture.

Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture addresses a hot research area and promises to set a standard for the field. Researchers and applied mathematicians interested in mathematical physics and relativity will find this work both fascinating and important.

Table of Contents

ELLIPTIC OPERATORS

Introduction

The Real and Complex Laplace Operators

Spinors

Spectral Resolutions

Manifolds with Boundary

Spectral Invariants

The Eta Invariant

Computing the Eta Invariant

DIFFERENTIAL GEOMETRY

Introduction

Riemannian Submersions

Characteristic Classes

The Geometry of Sphere and Principal Bundles

The Geometry of Circle Bundles

The Hopf Fibration

The Scalar Curvature

Levi-Civita and Spin Connections

POSITIVE CURVATURE

Introduction

Manifolds with Positive Ricci Curvature

Bordism and Connective K Theory

Calculations Involving the Eta Invariant

The Eta Invariant and Connective K Theory

Computing Connective K Theory Groups

SPECTRAL GEOMETRY OF RIEMANNIAN SUBMERSIONS

Introduction

Intertwining the Coderivitives

The Real Laplacian

The Complex Laplacian

The Spin Laplacian

Riemannian Submersions with Boundary

Heat Trace and Heat Content

Unresolved Questions

REFERENCES

Introduction

Main Bibliography

Bibliography of Harmonic Morphisms

Parabolic PDE Bibliography

NOTATION

INDEX

About the Series

Studies in Advanced Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT012000
MATHEMATICS / Geometry / General