Characteristic classes and differentiable manifolds

  • E. Thomas
Part of the C.I.M.E. Summer Schools book series (CIME, volume 41)


These lectures might be more accurately titled “The application of characteristic classes to geometric problems on manifolds.” In particular we will be interested in studying vector fields on manifolds. This first lecture gives the basic definitions we will need throughout the course.

1. Smooth manifolds. Let R denote the real numbers and Rn, for n ≥ 1, the space of n-tples (x 1,…,x n ), x i∈R. Let U be an open set in Rn. A map f: U → R will be called smooth if its partial derivatives of all orders exist and are continuous. More generally, a map f: U → Rq will be called smooth if each coordinate function f j is smooth, where f j: U → R is given by the following composition:


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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • E. Thomas
    • 1
  1. 1.National Science FoundationUSA

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