Submanifold: Difference between revisions

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[[Image:immersedsubmanifold_nonselfintersection.jpg|thumb|150px|Immersed submanifold open interval with interval ends mapped to arrow marked ends.]]
 
An '''immersed submanifold''' of a manifold ''M'' is athe subsetimage ''S'' togetherof with aan [[topologyimmersion (structuremathematics)|topologyimmersion]] and [[differential structure]] such that ''S'' is a manifold and the inclusion map ''i'' : ''SN'' → ''M''; isin angeneral [[injective]]this [[immersionimage will (mathematics)|immersion]]not be a submanifold as a subset, so the term is used loosely.
 
More narrowly, one can require that the map ''i'': ''N'' → ''M'' be an inclusion (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 ''i'' 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.