## 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

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 |

282 | |

283 | |

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algebraic allowable analytic apply associated base boundary calculation called chapter closed coefficients columns complex components condition connection consider constant construct continuous convenient coordinate system corresponding covariant cycles defined definition denote derivative determine differential dimensions effective effective cycles elements equal equations exists expressed follows functions fundamental geometry give given group manifold harmonic integrals Hence homologous importance independent ineffective intersection invariant invariant integrals Lemma linear metric multiplicity necessary neighbourhood non-singular null obtained orientation p-cycles p-fold integrals p-form parameters period matrix periods polynomial positive proof properties prove rank regular relations replace represents respect Riemannian manifold rows satisfies Similarly simple simplexes solution space substitution sufficient suppose tensor theorem theory transformation translation group unique variety vector write written zero