[Inquiry] Re: Differential And Riemannian Manifolds

[Jon Awbrey]

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DARM.  Note 7

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| 2.2.  Submanifolds, Immersions, Submersions
|
| Let X be a topological space, and Y a subset of X.
| We say that Y is "locally closed" in X if every point
| y in Y has an open neighborhood U in X such that Y |^| U
| is closed in U.  One verifies easily that a locally closed
| subset is the intersection of an open set and a closed set.
| For instance, any open subset of X is locally closed, and
| any open interval is locally closed in the plane.
|
| Let X be a manifold (of class C^p with p >= 0).  Let Y be a subset of X
| and assume that for each point y in Y there exists a chart (V, r) at y
| such that r gives an isomorphism of V with a product V_1 x V_2 where
| V_1 is open in some space E_1 and V_2 is open in some space E_2,
| and such that:
|
| r(Y |^| V) = V_1 x a_2
|
| for some point a_2 in V_2 (which we could take to be 0).  Then it is clear
| that Y is locally closed in X.  Furthermore, the map r induces a bijection:
|
| r_1  :  Y |^| V  ->  V_1.
|
| The collection of pairs (Y |^| V, r_1) obtained in the above manner constitues
| an atlas for Y, of class C^p.  The verification of this assertion, whose formal
| details we leave to the reader, depends on the following obvious fact.
|
| Lemma 2.1.  Let U_1, U_2, V_1, V_2 be open subsets of Banach spaces,
|
|             and g : U_1 x U_2 -> V_1 x V_2 a C^p-morphism.
|
|             Let a_2 be in U_2 and b_2 be in V_2
|
|             and assume that g maps U_1 x a_2 into V_1 x b_2.
|
|             Then the induced map:
|
|             g_1 : U_1 -> V_1
|
|             is also a morphism.
|
| Indeed, it is obtained as a composite map:
|
| U  ->  U_1 x U_2  ->  V_1 x V_2  ->  V_1,
|
| the first map being an inclusion and the third a projection.
|
| We have therefore defined a C^p-structure on Y which will be called
| a "submanifold" of X.  This structure satisfies a universal mapping
| property, which characterizes it, namely:
|
| | Given any map f : Z -> X from a manifold Z into X such that
| | f(Z) is contained in Y.  Let f_Y : Z -> Y be the induced map.
| | Then f is a morphism if and only if f_Y is a morphism.
|
| The proof of this assertion depends on Lemma 2.1, and is trivial.
|
| Finally, we note that the inclusion of Y into X is a morphism.
| 
| If Y is also a closed subspace of X, then
| we say that it is a "closed submanifold".
|
| Lang, DARM, pp. 23-24.
|
| Serge Lang,
|'Differential & Riemannian Manifolds',
| Springer-Verlag, New York, NY, 1995.

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