The Theory and Applications of Harmonic Integrals

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CUP Archive, May 25, 1989 - Mathematics - 284 pages
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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

Introduction
1
Manifolds of class u
6
The Biemannian metric
11
Orientation
13
Geometry of a Biemannian manifold
16
Differential geometry
17
Numerical tensors The metrical tensors
21
Parallel displacement
23
HARMONIC INTEGBALS 27 Definition of harmonic forms
107
Approximation by closed psetfl
113
Periods of harmonic integrals
117
Topology of algebraic manifolds II
168
Periods of harmonic integrals II
182
Complex parameters II
188
Properties of the period matrices of effective integrals
192
Change of metric li
198

Covariant differentiation
28
Geodesic coordinates
34
Topology
36
Complexes of olass v
49
Manifolds
50
Orientation
51
Duality
52
Intersections
54
Product manifolds
64
Multiple integrals
68
The theorem of Stokes
74
Calculus of forme
78
Periods
79
The first theorem of de Bham
87
Proof of de Rhams first theorem
92
De Rhams second theorem
100
Products of integrals and intersections of cycles
101
Some enumerative results
200
Defective systems of integrals 21
201
Applications to problems in algebraic geometry 2
212
Some results for surfaces 2
223
APPLICATIONS TO THE THEORY OF CONTINUOUS GROUPS
226
Continuous groups
227
Geometry of the transformation space
236
The transformation of tensors
240
Invariant integrals
243
The group manifold
259
The four main classes of simple groups 21
267
The orthogonal group Ou+l
275
The symplectic group Sty
279
Conclusion 21
280
Index
282
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