[Inquiry] Re: Differential And Riemannian Manifolds

[Jon Awbrey]

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DARM.  Note 13

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| 2.2.  Submanifolds, Immersions, Submersions (cont.)
|
| As with immersions and submersions,
| we have a characterization of
| transversal maps in terms
| of tangent spaces.
|
| Proposition 2.4.  Let X, Y be manifolds of class C^p (p >= 1)
|
|                   modeled on Banach spaces.
|
|                   Let f : X -> Y be a C^p-morphism,
|
|                   and W a submanifold of Y.
|
|                   The map f is transversal over W
|
|                   if and only if
|
|                   for each x in X such that f(x) lies in W,
|
|                   the composite map:
|
|                            T_x (f)
|                   T_x (X) ---------> T_w (Y) ---------> T_w (Y) / T_w (W),
|
|                   with w = f(x), is surjective and its kernel splits.
|
| Proof.  If f is transversal over W, then for each point x in X such
|         that f(x) lies in W, we choose charts as in the definition,
|         and reduce the question to one of maps of open subsets of
|         Banach spaces.  In that case, the conclusion concerning
|         the tangent spaces follows at once from the assumed
|         direct product decompositions.
|
|         Conversely, assume our condition on the tangent map.  The
|         question being local, we can assume that Y = V_1 x V_2 is a
|         product of open sets in Banach spaces such that W = V_1 x 0,
|         and we can also assume that X = U is open in some Banach space,
|         x = 0.  Then we let g : U -> V_2 be the map !p! o f, where !p!
|         is the projection, and note that our assumption means that
|         g'(0) is surjective and its kernel splits.  Furthermore,
|         g^(-1)(0) = f^(-1)(W).  We can then use Corollary 5.7
|         of the inverse mapping theorem to conclude the proof.
|
| Lang, DARM, pp. 27-28.
|
| Serge Lang,
|'Differential & Riemannian Manifolds',
| Springer-Verlag, New York, NY, 1995.

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