[Inquiry] Re: Differential And Riemannian Manifolds

[Jon Awbrey]

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DARM.  Note 9

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| 2.2.  Submanifolds, Immersions, Submersions (cont.)
|
| A morphism f : X -> Y will be called a "submersion" at a point x in X
| if there exists a chart (U, q) at x and a chart (V, r) at f(x) such that
| q gives an isomorphism of U on a product U_1 x U_2 (U_1 and U_2 open in
| some Banach spaces), and such that the map:
|
| rfq^-1  =  f_V,U  :  U_1 x U_2  ->  V
|
| is a projection.  One sees then that the image of a submersion is
| an open subset (a submersion is in fact an open mapping).  We say
| that f is a "submersion" if it is a submersion at every point.
|
| For manifolds modeled on Banach spaces,
| we have the usual criterion for immersions
| and submersions in terms of the derivative.
| 
| Proposition 2.2.  Let X, Y be manifolds of class C^p (p >= 1)
|
|                   modeled on Banach spaces.
|
|                   Let f : X -> Y be a C^p-morphism.
|
|                   Let x be in X.
|
|                   Then:
|
|                   1.  f is an immersion at x if and only if
|
|                       there exists a chart (U, q) at x and (V, r) at f(x)
|
|                       such that f'_V,U (qx) is injective and splits.
|
|                   2.  f is a submersion at x if and only if
|
|                       there exists a chart (U, q) at x and (V, r) at f(x)
|
|                       such that f'_V,U (qx) is surjective and its kernel splits.
|
| Proof.  This is an immediate consequence
|         of Corollaries 5.4 and 5.6 of
|         the inverse mapping theorem.
| 
| The conditions expressed in [Propositions 2.2.1 and 2.2.2] depend only on the
| derivative [f'], and if they hold for one choice of charts (U, q) and (V, r),
| respectively, then they hold for every choice of such charts.  It is therefore
| convenient to introduce a terminology in order to deal with such properties.
|
| Lang, DARM, p. 25.
|
| Serge Lang,
|'Differential & Riemannian Manifolds',
| Springer-Verlag, New York, NY, 1995.

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