Advertisement

Basics of Manifolds

  • Stephen Bruce Sontz
Chapter
Part of the Universitext book series (UTX)

Abstract

We start off with the idea of a chart. This is sometimes called a system of coordinates, but we feel, as does Lang (see [32]), that this both obscures the basic idea and impairs the recognition of an immediate generalization to Banach manifolds by introducing scads of unnecessary notation. The idea that Banach spaces provide an appropriate scenario for doing differential calculus goes back at least to the treatise [8] of Dieudonné. An extremely well-written text on smooth manifolds is Lee’s book [33].

Bibliography

  1. 8.
    J. Dieudonné, Foundations of Modern Analysis, Academic Press, 1960.Google Scholar
  2. 23.
    F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer, 1978 (reprinted 1995).Google Scholar
  3. 26.
    L. Hörmander, The Analysis of Linear Partial Operators I, Springer-Verlag, 1983.Google Scholar
  4. 27.
    D. Husemoller, Fibre Bundles, Graduate Texts in Mathematics, Vol. 20, Springer.Google Scholar
  5. 29.
    M. Kervaire, A manifold which does not admit any differentiable structure, Commun. Math. Helv. 34 (1960) 304–312.MathSciNetzbMATHGoogle Scholar
  6. 32.
    S. Lang, Fundamentals of Differential Geometry, Graduate Texts in Mathematics, Vol. 191, Springer, 1999.Google Scholar
  7. 33.
    J. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, Vol. 218, Springer, 2003.Google Scholar
  8. 37.
    J. Milnor, On manifolds homeomorphic to the 7-sphere, Ann. Math. 64 (1956) 399–405.MathSciNetCrossRefGoogle Scholar
  9. 43.
    S. Smale, Generalized Poincaré’s conjecture in dimensions greater than four, Ann. Math. 74 (1961) 391–406.MathSciNetCrossRefGoogle Scholar
  10. 47.
    R.S. Strichartz, The Way of Analysis, Jones and Bartlett, 2000.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Stephen Bruce Sontz
    • 1
  1. 1.Centro de Investigación en Matemáticas, A.C.GuanajuatoMexico

Personalised recommendations