[Inquiry] Re: Differential And Riemannian Manifolds

[Jon Awbrey]

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DARM.  Note 17

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| 2.2.  Submanifolds, Immersions, Submersions (cont.)
|
| In our category of manifolds, we shall deal only with cases
| when the fiber product can be taken to be the set-theoretic
| fiber product on which a manifold structure has been defined.
| (The set-theoretic fiber product is the set of pairs of points
| projecting on the same point.)  This determines the fiber product
| uniquely, and not only up to a unique isomorphism.
|
| Proposition 2.5.  Let f : X -> Z and g : Y -> Z
|
|                   be two C^p-morphisms with p >= 1.
|
|                   If they are transversal, then:
|
|                   (f x g)^(-1) (!D!_Z),
|
|                   together with the natural morphisms into X and Y
|
|                   (obtained from the projections),
|
|                   is a fiber product of f and g over Z.
|
| Proof.  Obvious.
|
| To construct a fiber product, it suffices to do it locally.
| Indeed, let f : X -> Z and g : Y -> Z be two morphisms.
| Let {V_i} be an open covering of Z, and let:
|
| f_i  :  f^(-1) (V_i)  ->  V_i
|
| and
|
| g_i  :  g^(-1) (V_i)  ->  V_i
|
| be the restrictions of f and g to the respective inverse images of V_i.
| Let P = (f x g)^(-1) (!D!_Z).  Then P consists of the points (x, y) with
| x in X and y in Y such that f(x) = g(y).  We view P as a subspace of X x Y
| (i.e. with the topology induced by that of X x Y).  Similarly, we construct
| P_i with f_i and g_i.  Then P_i is open in P.  The projections on the first
| and second factors give natural maps of P_i into f^(-1)(V_i) and g^(-1)(V_i),
| and of P into X and Y.
|
| Lang, DARM, p. 30.
|
| Serge Lang,
|'Differential & Riemannian Manifolds',
| Springer-Verlag, New York, NY, 1995.

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