# Differentiable Manifolds

## Forms, Currents, Harmonic Forms

• Georges de Rham
Book

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 266)

1. Front Matter
Pages I-X
2. Georges de Rham
Pages 1-2
3. Georges de Rham
Pages 3-14
4. Georges de Rham
Pages 15-33
5. Georges de Rham
Pages 34-78
6. Georges de Rham
Pages 79-98
7. Georges de Rham
Pages 99-160
8. Back Matter
Pages 161-167

### Introduction

In this work, I have attempted to give a coherent exposition of the theory of differential forms on a manifold and harmonic forms on a Riemannian space. The concept of a current, a notion so general that it includes as special cases both differential forms and chains, is the key to understanding how the homology properties of a manifold are immediately evident in the study of differential forms and of chains. The notion of distribution, introduced by L. Schwartz, motivated the precise definition adopted here. In our terminology, distributions are currents of degree zero, and a current can be considered as a differential form for which the coefficients are distributions. The works of L. Schwartz, in particular his beautiful book on the Theory of Distributions, have been a very great asset in the elaboration of this work. The reader however will not need to be familiar with these. Leaving aside the applications of the theory, I have restricted myself to considering theorems which to me seem essential and I have tried to present simple and complete of these, accessible to each reader having a minimum of mathematical proofs background. Outside of topics contained in all degree programs, the knowledge of the most elementary notions of general topology and tensor calculus and also, for the final chapter, that of the Fredholm theorem, would in principle be adequate.

### Keywords

Differenzierbare Mannigfaltigkeit Rham Riemannian manifold Varieties manifold

#### Authors and affiliations

• Georges de Rham
• 1
1. 1.LausanneSwitzerland

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-642-61752-2
• Copyright Information Springer-Verlag Berlin Heidelberg 1984
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-642-61754-6
• Online ISBN 978-3-642-61752-2
• Series Print ISSN 0072-7830
• Buy this book on publisher's site
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