[Inquiry] Re: Differential And Riemannian Manifolds

[Jon Awbrey]

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DARM.  Note 18

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| 2.2.  Submanifolds, Immersions, Submersions (concl.)
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| Proposition 2.6.  Assume that each P_i admits a manifold structure (compatible
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|                   with its topology) such that these maps are morphisms,
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|                   making P_i into a fiber product of f_i and g_i.
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|                   Then P, with its natural projections,
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|                   is a fiber product of f and g.
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| To prove the above assertion, we observe that the P_i form a covering of P.
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| Furthermore, the manifold structure on P_i |^| P_j induced by that of P_i or P_j
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| must be the same, because it is the unique fiber product structure over V_i |^| V_j,
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| for the maps f_ij and g_ij (defined on f^(-1)(V_i |^| V_j) and g^(-1)(V_i |^| V_j),
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| respectively).  Thus we can give P a manifold structure, in such a way that the
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| two projections into X and Y are morphisms, and make P into a fiber product
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| of f and g.
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| We shall apply the preceding discussion
| to vector bundles in the next chapter, and
| the following local criterion will be useful.
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| Proposition 2.7.  Let f : X -> Z be a morphism,
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|                   and g : Z x W -> Z be the
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|                   projection on the first factor.
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|                   Then f, g have a fiber product,
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|                   namely the product X x W
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|                   together with the morphisms
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|                   of the following diagram:
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|                 f x id
|   X x W o------------------>o Z x W
|         |                   |
|         |                   |
|         |                   |
|  proj_1 |                   | proj_2
|         |                   |
|         |                   |
|         v                   v
|       X o------------------>o Z
|                   f
|
| Lang, DARM, pp. 30-31.
|
| Serge Lang,
|'Differential & Riemannian Manifolds',
| Springer-Verlag, New York, NY, 1995.

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