The Theory and Applications of Harmonic Integrals

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CUP Archive, May 25, 1989 - Mathematics - 284 pages
First published in 1941, this book, by one of the foremost geometers of his day, rapidly became a classic. In its original form the book constituted a section of Hodge's essay for which the Adam's prize of 1936 was awarded, but the author substantially revised and rewrote it. The book begins with an exposition of the geometry of manifolds and the properties of integrals on manifolds. The remainder of the book is then concerned with the application of the theory of harmonic integrals to other branches of mathematics, particularly to algebraic varieties and to continuous groups. Differential geometers and workers in allied subjects will welcome this reissue both for its lucid account of the subject and for its historical value. For this paperback edition, Professor Sir Michael Atiyah has written a foreword that sets Hodges work in its historical context and relates it briefly to developments.
 

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Contents

RIEMANNIAN MANIFOLDS 1 Introduction
1
Manifolds of class
7
The Riemannian metric 4 Orientation
13
Geometry of a Riemannian manifold
16
Differential geometry 6 Tensors and their algebra
17
Numerical tonsors The metrical tensors 8 Parallel displacement
25
Covariant differentiation
28
Riemannian geometry 11 Geodesic coordinates
34
De Rhams second theorem
143
The equations satisfied by a harmonic tensor
144
APPLICATIONS TO ALGEBRAIC VARIETIES
147
Algebraic varieties
148
Construction of the Riemannian manifold
150
Discussion of the metric
154
The affine connection and curvature tensor
159
Harmonic integrals on an algebraic manifold
165

Polyhedral complexes
36
Complexes of class v
45
Manifolds
50
Orientation
51
Duality
54
Intersections
55
Product manifolds
65
INTEGRALS AND THEIR PERIODS
68
Multiple integrals
69
The theorem of Stokes
77
Calculus of forme
78
Periods
80
The first theorem of de Rham
91
Proof of de Rhams first theorem
95
De Rhams second theorem
100
Products of integrals and intersections of cycles 100
101
HARMONIC INTEGRALS
107
Definition of harmonic forms
109
Approximation by closed psets
115
Periods of harmonic integrals
117
preliminary considerations page
119
The existence theorem continued
130
Digression on the solution of integral equations
134
The existence theorem concluded
139
The fundamental forms
168
An analysis of forms associated with an algebraic manifold
171
The classification of harmonic integrals on an algebraic manifold
178
Topology of algebraic manifolds
182
Periods of harmonic integrals
185
Complex parameters
188
Properties of the period matrices of effective integrals
192
Change of metric
198
Some enumerative results
200
Defective systems of integrals
201
Applications to problems in algebraic geometry
212
Some results for surfaces
218
APPLICATIONS TO THE THEORY OF CONTINUOUS GROUPS 63 Continuous groups
226
Geometry of the transformation space 236 54 Geometry of the transformation space 55 The transformation of tensors
240
Invariant integrals
242
The group manifold 249 57 The group manifold 68 The four main classes of simple groups
258
The unimodular group Ln
264
The orthogonal group Oxy+1
272
The orthogonal group Oy
279
Conclusion
280
107
282
117
283
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