Differentiable manifolds: forms, currents, harmonic forms
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 definitio…
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- Full title :Differentiable manifolds : forms, currents, harmonic forms / Georges de Rham.
Summary
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.
Read More ↓Edition Statements
- Publication Date:©1984
Publication Statements
- Publisher(s):Springer-Verlag
- Place of Publication:Berlin, Heidelberg
- Language:eng
- Publication Date:©1984
The Physical Object
- Description:1 online resource
Edition Identifiers
- ISBN:9783642617522, 3642617522
- DOI:10.1007/978-3-642-61752-2
- oclc:ocn840293679
- LC Classification:QA614.3
- Dewey Decimal:514/.34
Table of Contents
I. Notions About Manifolds -- ʹ1. The Notion of a Manifold and a Differentiable Structure -- ʹ2. Partition of Unity. Functions on Product Spaces -- ʹ3. Maps and Imbeddings of Manifolds -- II. Differential Forms -- ʹ4. Differential Forms of Even Type -- ʹ5. Differential Forms of Odd Type. Orientation of Manifolds and Maps -- ʹ6. Chains. Stokes' Formula -- ʹ7. Double Forms -- III. Currents -- ʹ8. Definition of Currents -- ʹ9. The Vector Spaces E, D, Ep, and Dp -- ʹ10. The Vector Spaces D́, É, D́p, and Ép -- ʹ11. Boundary of a Current. Image of a Current by a Map -- ʹ12. Double Currents -- ʹ13. Transformations of Double Forms and Currents by a Map -- ʹ14. Homotopy Formulas -- ʹ15. Regularization -- ʹ16. Operators Associated with a Double Current -- ʹ17. Reflexitivity of E and D. Regular Operators and Regularizing Operators -- IV. Homologies -- ʹ18. Homology Groups -- ʹ19. Homologies in IRn -- ʹ20. The Kronecker Index -- ʹ21. Homologies Between Forms and Chains in a Manifold Endowed with a Polyhedral Subdivision -- ʹ22. Duality in a Manifold Endowed with a Polyhedral Subdivision -- ʹ23. Duality in Any Differentiable Manifold -- V. Harmonic Forms -- ʹ24. Riemannian Spaces. Adjoint Form -…
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