Abstract
1. DIFFERENTIABLE MANIFOLDS. A differentiable manifold, Xn, is an astract object having the following properties : (1) It is a topological manifold, covered with open sets Ui. It is usually assumed to be paracompact. In most of these lectures we assume it to be compact (2) There is a map : ϕi : Ui.→ En for each Ui. These establish coordinates in Ui. (3) In overlapping open sets, i.e. in Ui∩Uj, the corresponding coordinates are related by differentiable functions.
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