Submanifold: Difference between revisions

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An '''immersed submanifold''' of a manifold ''M'' is the image ''S'' of an [[immersion (mathematics)|immersion]] map ''f'': ''N'' → ''M''; in general this image will not be a submanifold as a subset, and an immersion map need not even be [[injective]] (one-to-one) – it can have self-intersections.<ref>{{harvnb|Sharpe|1997|page=26}}.</ref>
 
More narrowly, one can require that the map ''f'': ''N'' → ''M'' be an inclusioninjection (one-to-one), in which we call it an [[injective]] [[immersion (mathematics)|immersion]], and define an '''immersed submanifold''' to be the image subset ''S'' together with a [[topology (structure)|topology]] and [[differential structure]] such that ''S'' is a manifold and the inclusion ''f'' is a [[diffeomorphism]]: this is just the topology on ''N,'' which in general will not agree with the subset topology: in general the subset ''S'' is not a submanifold of ''M,'' in the subset topology.
 
Given any injective immersion ''f'' : ''N'' → ''M'' the [[image (mathematics)|image]] of ''N'' in ''M'' can be uniquely given the structure of an immersed submanifold so that ''f'' : ''N'' → ''f''(''N'') is a [[diffeomorphism]]. It follows that immersed submanifolds are precisely the images of injective immersions.
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