Recent developments in the field of differential geometry have been so extensive that a new book with particular emphasis on current work in Riemannian geometry is clearly necessary. This new text brilliantly serves that purpose and includes an elementary account of twistor spaces that will interest both applied mathematicians and physicists. It presents recent developments in the theory of harmonic spaces, commutative spaces, and mean-value theorems previously only available in the source literature. The final chapter provides the only account available in book form of manifolds known as `Willmore surfaces', illustrated by a series of computer-generated pictures. This book is sure to be welcomed by researchers, teachers, and students interested in the latest developments in differential geometry.
Author(s): T. J. Willmore
Series: Oxford Science Publications
Publisher: Clarendon
Year: 1993
Language: English
Pages: 336
City: Oxford
Cover
Title page
Date-line
Dedication
Preface
Contents
1 Differentiable manifolds
1.1 Preliminaries
1.2 Differentiable manifolds
1.3 Tangent vectors
1.4 The differential of a map
1.5 The tangent and cotangent bundles
1.6 Submanifolds; the inverse and implicit function theorems
1.7 Vector fields
1.8 Distributions
1.9 Exercises
2 Tensors and differential forms
2.1 Tensor product
2.2 Tensor fields and differential forms
2.3 Exterior derivation
2.4 Differential ideals and the theorem of Frobenius
2.5 Orientation of manifolds
2.6 Covariant differentiation
2.7 Identities satisfied by the curvature and torsion tensors
2.8 The Koszul connexion
2.9 Connexions and moving frames
2.10 Tensorial forms
2.11 Exercises
3 Riemannian manifolds
3.1 Riemannian metrics
3.2 Identities satisfied by the curvature tensor of a Riemannian manifold
3.3 Sectional curvature
3.4 Geodesies
3.5 Normal coordinates
3.6 Volume form
3.7 The Lie derivative
3.8 The Hodge star operator
3.9 Maxwell's equations
3.10 Consequences of the theorems of Stokes and of Hodge
3.11 The space of curvature tensors
3.12 Conformal metrics and conformal connexions
3.13 Exercises
4 Submanifold theory
4.1 Submanifolds
4.2 The equations of Gauss, Codazzi, and Ricci
4.3 The method of moving frames
4.4 The theory of curves
4.5 The theory of surfaces
4.6 Surface theory in classical notation
4.7 The Gauss-Bonnet theorem
4.8 Exercises
5 Complex and almost-complex manifolds
5.1 Complex structures on vector spaces
5.2 Hermitian metrics
5.3 Almost-complex manifolds
5.4 Walker derivation
5.5 Hermitian metrics and Kaehler metrics
5.6 Complex manifolds
5.7 Kaehler metrics in local coordinate systems
5.8 Holomorphic sectional curvature
5.9 Curvature and Chern forms of Kaehler manifolds
5.10 Review of complex structures on vector spaces
5.11 The twistor space of a Riemannian manifold
5.12 The twistor space of $S^4$
5.13 Twistor spaces of spheres
5.14 Riemannian immersions and Codazzi's equation
5.15 Isotropic immersions
5.16 The bundle of oriented orthonormal frames
5.17 The relationship between the frame bundle and the twistor space
5.18 Integrability of the almost-complex structures on the twistor space
5.19 Interpretation of the integrability condition
5.20 Exercises
6 Special Riemannian manifolds
6.1 Harmonic and related metrics
6.2 Normal coordinates
6.3 The distance function $Omega$
6.4 The discriminant function
6.5 Geodesic polar coordinates
6.6 Jacobi fields
6.7 Definition of harmonic metrics
6.8 Curvature conditions for harmonic manifolds
6.9 Consequences of the curvature conditions
6.10 Harmonic manifolds of dimension 4
6.11 Mean-value properties
6.12 Commutative metrics
6.13 The curvature of Einstein symmetric spaces
6.14 D'Atri spaces
6.15 Exercises
7 Special Riemannian submanifolds
7.1 Introduction
7.2 Surfaces in $E^3$
7.3 Conformal invariance
7.4 The Euler equation
7.5 Willmore surfaces and minimal submanifolds of $S^3$
7.6 Pinkall's inequality
7.7 Related work by Robert L. Bryant
7.8 The conformal invariants of Li and Yau, and of Gromov
7.9 Contributions from the German school and their colleagues
7.10 Other recent contributions to the subject
7.11 Exercises
Appendix: Partitions of unity
Bibliography
Index
