Abstract
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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References
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Thomas, E. (2010). Characteristic classes and differentiable manifolds. In: Martinelli, E. (eds) Classi caratteristiche e questioni connesse. C.I.M.E. Summer Schools, vol 41. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11048-1_4
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