Differentiable Manifolds

I. Notions About Manifolds.- §1. The Notion of a Manifold and a Differentiable Structure.- §2. Partition of Unity. Functions on Product Spaces.- §3. Maps and Imbeddings of Manifolds.- II. Differential Forms.- §4. Differential Forms of Even Type.- §5. Differential Forms of Odd Type. Orientation of Manifolds and Maps.- §6. Chains. Stokes' Formula.- §7. Double Forms.- III. Currents.- §8. Definition of Currents.- §9. The Vector Spaces E, D, Ep, and Dp.- §10. The Vector Spaces D´, E´, D´p, and E´p.- §11. Boundary of a Current. Image of a Current by a Map.- §12. Double Currents.- §13. Transformations of Double Forms and Currents by a Map.- §14. Homotopy Formulas.- §15. Regularization.- §16. Operators Associated with a Double Current.- §17. Reflexitivity of E and D. Regular Operators and Regularizing Operators.- IV. Homologies.- §18. Homology Groups.- §19. Homologies in IRn.- §20. The Kronecker Index.- §21. Homologies Between Forms and Chains in a Manifold Endowed with a Polyhedral Subdivision.- §22. Duality in a Manifold Endowed with a Polyhedral Subdivision.- §23. Duality in Any Differentiable Manifold.- V. Harmonic Forms.- §24. Riemannian Spaces. Adjoint Form.- §25. The Metric Transpose of an Operator. The Operators ? and ?.- §26. Expressions of the Operators d, ?, and ? Using Covariant Derivatives.- §27. Properties of the Geodesic Distance.- §28. The Parametrix.- §29. The Regularity of Harmonic Currents.- §30. The Local Study of the Equation ??= ?. Elementary Kernels.- §31. The Equation ?S = T on a Compact Space. The Operators H and G.- §32. The Decomposition Formula in a Non-Compact Space.- §33. Explicit Formula for the Kronecker Index.- §34. The Analyticity of Harmonic Forms.- §35. Square Summable Harmonic Forms on aComplete Riemannian Space.- List of Notation.