|WikiProject Mathematics||(Rated Start-class, Low-importance)|
The top diagram is incorrect - an immersed submanifold is the image of an injective immersion, and hence cannot have "self-intersections". --220.127.116.11 19:23, 11 September 2007
- It's true that the image is in conflict with the definition given. In my experience, people often use the words "immersed submanifold" to refer to submanifolds with self-intersections, so I vote for changing the definition. Jim 21:39, 11 September 2007 (UTC)
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.)
- In the first statement, what does "the structure of an immersed submanifold" even mean? By the definition in the article, an immersed submanifold is the image of an immersion, so the only "structure" it has is the subspace topology inherited from M. In particular, this entails no particular smooth structure, so it's meaningless to say that in f(N)'s "immersed submanifold structure" some map onto f(N) is a diffeomorphism.
- Assuming therefore "the structure of an immersed submanifold" means just the subspace topology from M, the first statement would be trying to say: "Given any injective immersion f : N → M, equipping the image f(N) of N in M with the subspace topology, the map f : N → f(N) is a homeomorphism." This however is exactly false. See later comments (in the same article!) on the distinction between injective immersions and embeddings.
- Applying an earlier definition, the second statement says: "images of immersions are precisely images of injective immersions." I don't think this is true -- for instance, the union of and in the unit torus is the image of an immersion of , but feels like it shouldn't be the image of an injective immersion.
I propose deleting the paragraph.
- 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? --18.104.22.168 (talk) 20:40, 3 February 2016 (UTC)