ONT Re: Differential And Riemannian Manifolds
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DARM. Note 14
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| 2.2. Submanifolds, Immersions, Submersions (cont.)
|
| If E is a Banach space, then the diagonal !D! in E x E
| is a closed subspace and splits: Either factor E x 0
| or 0 x E is a closed complement. Consequently, the
| diagonal is a closed submanifold of E x E. If X
| is any manifold of class C^p, p >= 1, then the
| diagonal is therefore also a submanifold.
| (It is closed of course if and only if
| X is Hausdorff.)
|
| Let f : X -> Z and g : Y -> Z be two C^p-morphisms, p >= 1.
| We say that they are "transversal" if the morphism:
|
| f x g : X x Y -> Z x Z
|
| is transversal over the diagonal. We remark right away
| that the surjectivity of the map in Proposition 2.4 can
| be expressed in two ways. Given two points x in X and
| y in Y such that f(x) = g(y) = z, the condition:
|
| Im (T_x f) + Im (T_y g) = T_z (Z)
|
| is equivalent to the condition:
|
| Im (T_(x,y) (f x g)) + T_(z,z) (!D!) = T_(z,z) (Z x Z).
|
| Thus in the finite dimensional case, we could
| take it as the definition of transversality.
|
| Lang, DARM, pp. 28-29.
|
| Serge Lang,
|'Differential & Riemannian Manifolds',
| Springer-Verlag, New York, NY, 1995.
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