[Inquiry] Re: Differential And Riemannian Manifolds

[Jon Awbrey]

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DARM.  Note 11

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We now define the "derivative" or the "differential" of a map.
The "derivative" is Lang's name for what most mathematicians
call the "differential", and vice versa, so the reader may
take it as an object lesson in differential translation.

| 2.2.  Submanifolds, Immersions, Submersions (cont.)
|
| If U, V are open in Banach spaces, then to every morphism f
| of class C^p (p >= 1) we can associate its derivative Df(x).
| If now f : X -> Y is a morphism of one manifold into another,
| and x a point of X, then by means of charts we can interpret
| the derivative of f on each chart at x as a mapping:
|
| df(x)  =  T_x f  :  T_x (X)  ->  T_f(x) (Y).
|
| Indeed, this map T_x f is the unique linear map having the following property.
| If (U, q) is a chart at x and (V, r) is a chart at f(x) such that f(U) c V
| and ^v^ is a tangent vector at x represented by v in the chart (U, q),
| then:
|
| T_x f(^v^)
|
| is the tangent vector at f(x) represented by (Df_V,U (x))v.
| The representation of T_x f on the spaces of charts can be
| given in the form of a diagram:
|
|       T_x (X)  o-------->o  E
|                |         |
|         T_x f  |         |  f'_V,U (x)
|                v         v
|    T_f(x) (Y)  o-------->o  F
|
| [NB.  f'_V,U (x) = Df_V,U (x), as an alternate notation.]
|
| The map T_x f is obviously continuous and linear
| for the structure of topological vector space
| which we have placed on T_x (X) and T_f(x) (Y).
|
| As a matter of notation, we shall sometimes write f_*,x instead of T_x f.
|
| The operation T satisfies an obvious functorial property,
| namely, if f : X -> Y and g : Y -> Z are morphisms, then:
|
| T_x (g o f)  =  (T_f(x) g) o (T_x f).
|
| T_x (id)     =  id.
|
| We may reformulate Proposition 2.2:
|
| Proposition 2.3.  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
|
|                       the map T_x f is injective and splits.
|
|                   2.  f is a submersion at x if and only if
|
|                       the map T_x f is surjective and its kernel splits.
|
| Note.  If X, Y are finite dimensional, then the condition that T_x f splits
| is superfluous.  Every subspace of a finite dimensional vector space splits.
|
| Lang, DARM, pp. 26-27.
|
| Serge Lang,
|'Differential & Riemannian Manifolds',
| Springer-Verlag, New York, NY, 1995.

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