ONT Re: Differential And Riemannian Manifolds
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DARM. Note 4
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| 2.1. Atlases, Charts, Morphisms (cont.)
|
| The collection of C^p-manifolds will be denoted by "Man^p".
| If we look only at those modeled on spaces in a category $A$
| then we write "Man^p ($A$)". Those modeled on a fixed E will
| be denoted by "Man^p (E)". We shall make these into categories
| by defining morphisms below.
|
| Let X be manifold, and U an open subset of X. Then it is possible,
| in the obvious way, to induce a manifold structure on U, by taking
| as charts the intersections:
|
| (U_i |^| U, q_i | (U_i |^| U)).
|
| [NB. "f | S" indicates the function f as restricted to the set S.]
|
| If X is a topological space, covered by open subsets V_j, and if we are
| given on each V_j a manifold structure such that for each pair j, j' the
| induced structure on V_j |^| V_j' coincides, then it is clear that we can
| give to X a unique manifold structure inducing the given ones on each V_j.
|
| Lang, DARM, pp. 21-22.
|
| Serge Lang,
|'Differential & Riemannian Manifolds',
| Springer-Verlag, New York, NY, 1995.
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