ONT Differential And Riemannian Manifolds
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DARM. Note 1
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| Excerpts from 'Differential And Riemannian Manifolds' by Serge Lang
|
| Chapt 2. Manifolds
|
| Starting with open subsets of Banach spaces [take R^n as a typical example],
| one can glue them together with C^p-isomorphisms [bijective mappings that
| are continuously differentiable up to order at least p]. The result is
| called a manifold. We begin by giving the formal definition. We then
| make manifolds into a category, and discuss special types of morphisms.
| We define the tangent space at each point, and apply the criteria
| following the inverse function theorem to get a local splitting
| of a manifold when the tangent space splits at a point.
|
| We shall wait until the next chapter to give
| a manifold structure to the union of all the
| tangent spaces.
|
| 2.1. Atlases, Charts, Morphisms
|
| Let X be a set. An "atlas" of class C^p (p >= 0) on X
| is a collection of pairs (U_i, q_i) (i ranging in some
| indexing set), satisfying the following conditions:
|
| AT 1. Each U_i is a subset of X and the U_i cover X.
|
| AT 2. Each q_i is a bijection of U_i onto an open subset q_i (U_i) of
some Banach space E_i, and for any i, j, [we have the fact that]
| q_i (U_i |^| U_j) is open in E_i.
|
| AT 3. The map
|
| (q_j) o (q_i)^(-1) : q_i (U_i |^| U_j) -> q_j (U_i |^| U_j)
|
| is a C^p-isomorphism for each pair of indices i, j.
|
| It is a trivial exercise in point set topology to prove that one
| can give X a topology in a unique way such that each U_i is open,
| and the q_i are topological isomorphisms.
|
| Lang, DARM, pp. 20-21.
|
| Serge Lang,
|'Differential And Riemannian Manifolds' (DARM),
| Springer-Verlag, New York, NY, 1995, pp. 20-21.
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