## The Theory and Applications of Harmonic IntegralsFirst 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 |

282 | |

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### Common terms and phrases

affine connection algebraic variety allowable coordinate systems analytic Betti number boundary called cell chain chapter closed form complex components consider contravariant vector convenient coordinate system xv correspondence covariant vector cycles rp define definition denote derivative dimensions domain dual dxil Euclidean space exists finite follows functions given group manifold harmonic form harmonic integrals Hence homeomorphism homologous independent infinitesimal intersection matrix invariant integrals lemma linear locus M(Vp manifold of class metrical tensor necessary and sufficient neighbourhood non-singular matrix null form number space obtained orientation of Ep oriented simplex p-chain p-cycles p-fold integral p-form p-set p-simplex parallel displacement parameters period matrix periods zero polynomial properties prove real numbers relations Rham's Riemann surface Riemannian geometry Riemannian manifold Riemannian metric Riemannian space satisfies the equation simplex Ep skew-symmetric solution sufficient condition suppose theorem theory topological torsion coefficients transformation translation group unique write