ONT Re: Differential And Riemannian Manifolds
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DARM. Note 8
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| 2.2. Submanifolds, Immersions, Submersions (cont.)
|
| Suppose that X is finite dimensional of dimension n, and that Y
| is a submanifold of dimension m. Then from the definition we see
| that the local product structure in the neighborhood of a point of
| Y can be expressed in terms of local coordinates as follows. Each
| point P of Y has an open neighborhood U in X with local coordinates
| (x_1, ..., x_n) such that the points of Y in U are precisely those
| whose last n - m coordinates are 0, that is, those points having
| coordinates of type:
|
| (x_1, ..., x_m, 0, ..., 0).
|
| Let f : Z -> X be a morphism, and let z be in Z. We shall say that f is
| an "immersion" at z if there exists an open neighborhood Z_1 of z in Z
| such that the restriction of f to Z_1 induces an isomorphism of Z_1
| onto a submanifold of X. We say that f is an "immersion" if it is
| an immersion at every point.
|
| Notice that there exist injective immersions
| which are not isomorphisms onto submanifolds,
| as given by the following example:
| ________
| / \
| / \
| | |
| | |
| \ V
| \__________________________________________
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| (The arrow means that the line approaches itself without touching.)
|
| An immersion which does give an isomorphism onto a submanifold is
| called an "embedding", and it is called a "closed embedding" if
| this submanifold is closed.
|
| Lang, DARM, pp. 24-25.
|
| Serge Lang,
|'Differential & Riemannian Manifolds',
| Springer-Verlag, New York, NY, 1995.
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