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The top diagram is incorrect - an immersed submanifold is the image of an injective immersion, and hence cannot have "self-intersections". -- 18.87.1.187 19:23, 11 September 2007
The paragraph,
Given any injective immersion f : N → M the image of N in M can be uniquely given the structure of an immersed submanifold so that f : N → M is a diffeomorphism. It follows that immersed submanifolds are precisely the images of injective immersions.
occurs in the section on immersed submanifolds. I think both statements are false. (Similar statements are true for embedded submanifolds, however, which might be where the confusion's coming from.)
I propose deleting the paragraph.
131.111.213.32 ( talk) 02:05, 25 January 2010 (UTC)
It is very confusing having the diagram of a self-intersecting line displayed alongside the lead. It may be appropriate alongside the section on "immersion submanifold" (and, even then, only provided the caption clarify whether or not it is also a submanifold), but only an unambiguous/actual submanifold should be depicted alongside the lead. Also, the use of the word "straight" in the caption is confusing (it doesn't look like a straight line, is it a mistake or a reference to another space?). Maybe the word " curve" is more appropriate, but maybe technically not: it is the image of many possible paths (maps from R to R2), but (since it selfintersects) is it still strictly 1D? Cesiumfrog ( talk) 01:26, 3 May 2011 (UTC)
The intrinsic definition of embedded submanifold seems to imply that every embedded submanifold is locally flat. But to my best knowledge, this is not the case in general. Can somebody clarify this? -- 131.212.251.119 ( talk) 20:40, 3 February 2016 (UTC)
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The top diagram is incorrect - an immersed submanifold is the image of an injective immersion, and hence cannot have "self-intersections". -- 18.87.1.187 19:23, 11 September 2007
The paragraph,
Given any injective immersion f : N → M the image of N in M can be uniquely given the structure of an immersed submanifold so that f : N → M is a diffeomorphism. It follows that immersed submanifolds are precisely the images of injective immersions.
occurs in the section on immersed submanifolds. I think both statements are false. (Similar statements are true for embedded submanifolds, however, which might be where the confusion's coming from.)
I propose deleting the paragraph.
131.111.213.32 ( talk) 02:05, 25 January 2010 (UTC)
It is very confusing having the diagram of a self-intersecting line displayed alongside the lead. It may be appropriate alongside the section on "immersion submanifold" (and, even then, only provided the caption clarify whether or not it is also a submanifold), but only an unambiguous/actual submanifold should be depicted alongside the lead. Also, the use of the word "straight" in the caption is confusing (it doesn't look like a straight line, is it a mistake or a reference to another space?). Maybe the word " curve" is more appropriate, but maybe technically not: it is the image of many possible paths (maps from R to R2), but (since it selfintersects) is it still strictly 1D? Cesiumfrog ( talk) 01:26, 3 May 2011 (UTC)
The intrinsic definition of embedded submanifold seems to imply that every embedded submanifold is locally flat. But to my best knowledge, this is not the case in general. Can somebody clarify this? -- 131.212.251.119 ( talk) 20:40, 3 February 2016 (UTC)