Smooth Manifolds
A program of the course at the 1st Italian Diffiety School,
(Forino, July 17  - August 1, 1998)
and at the 1st Russian Diffiety School
(Pereslavl-Zalessky, January 26 - February 5, 1998)

  1. Algebraic approach to manifolds. Interpretation of functions as measuring means.
  2. Spectrum of a K-algebra. Zarissky topology. Maps of spectrums. Examples.
  3. The spectrum theorem.
  4. Smooth hull of a R-algebra. Examples.
  5. Algebraic definition of smooth manifolds.
  6. Manifolds with boundary.
  7. Submanifolds, quotient manifolds and Cartesian product of manifolds.
  8. Coordinate definition of smooth manifolds.
  9. Examples. Configuration manifolds.
  10. Smooth maps.
  11. Equivalence of two definitions of manifolds.
  12. Diffeomorphisms, embeddings, immersions. The Sard theorem.
  13. Tangent vectors as differentiations at a point.
  14. Vector fields as differentiations of algebras.
  15. Restrictions of vector fields to submanifolds and differentiations from algebras to modules.
  16. Tangent space. Tangent map. Tangent bundle.
  17. Bundles. Product bundles. Vector bundles. Subbundles. Modules of sections.
  18. Examples. Principle bundles. Shtifel manifolds. Grassmann manifolds.
  19. Morphisms of bundles. The category of bundles.
  20. Whitney product of bundles and tensor product of modules of sections. Examples.
  21. Restrictions of bundles and induced bundles.
  22. Connectedness in a principle bundle. Form of connectedness.
  23. Cotangent bundle. Vector fields and differential forms as sections. Tensor bundles.
  24. Commutator of vector fields. Lie algebras of vector fields.
  25. Phase flow of a vector field. Lie derivative.

What to read:
  1. Sternberg, S.: Lectures on Differential Geometry, Prentice Hall, Inc. Englewood Cliffs, New York, 1964, chap. 3, par. 5.
  2. Bocharov, A.V., Verbovetsky, A.M., Vinogradov, A.M., and other: Symmetries and conservation laws of mathematical physics equations, Faktorial, Moscow, 1997 (in Russian) and AMS, 1999 (in English), chap. 1, par. 3.

Questions and suggestions should go to J. S. Krasil'shchik, josephk @