Author	Rham, Georges de. author
Title	Differentiable Manifolds [electronic resource] : Forms, Currents, Harmonic Forms / by Georges de Rham
Imprint	Berlin, Heidelberg : Springer Berlin Heidelberg, 1984
Connect to	http://dx.doi.org/10.1007/978-3-642-61752-2
Descript	X, 170 p. online resource

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

I. Notions About Manifolds -- ยง1. The Notion of a Manifold and a Differentiable Structure -- ยง2. Partition of Unity. Functions on Product Spaces -- ยง3. Maps and Imbeddings of Manifolds -- II. Differential Forms -- ยง4. Differential Forms of Even Type -- ยง5. Differential Forms of Odd Type. Orientation of Manifolds and Maps -- ยง6. Chains. Stokesโ{128}{153} Formula -- ยง7. Double Forms -- III. Currents -- ยง8. Definition of Currents -- ยง9. The Vector Spaces E, D, Ep, and Dp -- ยง10. The Vector Spaces Dยด, Eยด, Dยดp, and Eยดp -- ยง11. Boundary of a Current. Image of a Current by a Map -- ยง12. Double Currents -- ยง13. Transformations of Double Forms and Currents by a Map -- ยง14. Homotopy Formulas -- ยง15. Regularization -- ยง16. Operators Associated with a Double Current -- ยง17. Reflexitivity of E and D. Regular Operators and Regularizing Operators -- IV. Homologies -- ยง18. Homology Groups -- ยง19. Homologies in IRn -- ยง20. The Kronecker Index -- ยง21. Homologies Between Forms and Chains in a Manifold Endowed with a Polyhedral Subdivision -- ยง22. Duality in a Manifold Endowed with a Polyhedral Subdivision -- ยง23. Duality in Any Differentiable Manifold -- V. Harmonic Forms -- ยง24. Riemannian Spaces. Adjoint Form -- ยง25. The Metric Transpose of an Operator. The Operators ? and ? -- ยง26. Expressions of the Operators d, ?, and ? Using Covariant Derivatives -- ยง27. Properties of the Geodesic Distance -- ยง28. The Parametrix -- ยง29. The Regularity of Harmonic Currents -- ยง30. The Local Study of the Equation ??= ?. Elementary Kernels -- ยง31. The Equation ?S = T on a Compact Space. The Operators H and G -- ยง32. The Decomposition Formula in a Non-Compact Space -- ยง33. Explicit Formula for the Kronecker Index -- ยง34. The Analyticity of Harmonic Forms -- ยง35. Square Summable Harmonic Forms on a Complete Riemannian Space -- List of Notation