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| ``Re: inclusion map on Riemannian manifolds''
by VR on 2006-04-14 08:22:32 |
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| In your 2nd definition, it should be a collection of "slice" charts covering M, because if the submanifold is compact, then of course, no single chart (of N) can cover N.
The definition of (smooth) embedding I use is : injective immersion + topological embedding.
Defn. (1) => Defn. (2) :
Let (G, g) be a generic slice chart for M. An atlas for N is given by the collection {pg : G intersect N --> IR^m}, where p : IR^n --> IR^m is the canonical projection. With this smooth structure, it is straightforward to show that i : N --> M is a topological embedding and an injective immersion (in fact, it can also be shown that the smooth structure so obtained is the unique one making i a smooth embedding).
Defn. (2) => Defn. (1) :
Since i : N --> M is a a smooth embedding, the rank theorem can be used to obtain charts u : U subset IR^m --> V subset IR^n of the form (x_1, ..., x_m) --> (x_1, ..., x_m, 0, ..., 0). These charts can, after possibly some shrinking of domain, be used to easily obtain slice charts for N.
Details : see John M. Lee, Introduction to Smooth Manifolds (Springer 2003). |
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