compact manifold
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From: Maximilian Rogers on 9 Jan 2010 11:44 On Jan 9, 6:57 am, William Elliot <ma...(a)rdrop.remove.com> wrote: > On Sat, 9 Jan 2010, Maximilian Rogers wrote: > > On Jan 9, 2:19 am, William Elliot <ma...(a)rdrop.remove.com> wrote: > >> On Fri, 8 Jan 2010, Maximilian Rogers wrote: > >>> How can I show that a compact n- manifold does not embed in R^n? > > >> Does the unit sphere S^2, embed into R^2? > > > I know that the sphere doesn't embed in R^2, I wouldn't necessarily > > know hoe to prove it, though... > > Does the unit circle S^1, embed into R^1 ? I believe not, unless you remove the point at infinity, right?
From: William Elliot on 9 Jan 2010 23:10 On Sat, 9 Jan 2010, Maximilian Rogers wrote: >>>>> How can I show that a compact n- manifold does not embed in R^n? >> >>>> Does the unit sphere S^2, embed into R^2? >> >>> I know that the sphere doesn't embed in R^2, I wouldn't necessarily >>> know hoe to prove it, though... >> >> Does the unit circle S^1, embed into R^1 ? > > I believe not, unless you remove the point at infinity, right? > What point at infinity?
From: David C. Ullrich on 10 Jan 2010 07:15 On Sat, 9 Jan 2010 03:34:01 -0800 (PST), Maximilian Rogers <max.rogers123(a)gmail.com> wrote: >On Jan 9, 2:19�am, William Elliot <ma...(a)rdrop.remove.com> wrote: >> On Fri, 8 Jan 2010, Maximilian Rogers wrote: >> > How can I show that a compact n- manifold does not embed in R^n? >> >> Does the unit sphere S^2, embed into R^2? > >I know that the sphere doesn't embed in R^2, I wouldn't necessarily >know hoe to prove it, though... > >The invariance of dimension theorem says that R^m=R^n iff m=n, right? That's one very special case of the invariance of domain theorem. >I am not sure how to use it to prove what i want..could you give me >more details, please? First you need to find out what the theorem actually says. I could just tell you, but since you evidently have access to the internet it would probably be more instructive for you to simply look it up. On wikipedia for example.
From: Maximilian Rogers on 10 Jan 2010 18:23 On Jan 10, 6:15 am, David C. Ullrich <ullr...(a)math.okstate.edu> wrote: > On Sat, 9 Jan 2010 03:34:01 -0800 (PST), Maximilian Rogers > > <max.rogers...(a)gmail.com> wrote: > >On Jan 9, 2:19 am, William Elliot <ma...(a)rdrop.remove.com> wrote: > >> On Fri, 8 Jan 2010, Maximilian Rogers wrote: > >> > How can I show that a compact n- manifold does not embed in R^n? > > >> Does the unit sphere S^2, embed into R^2? > > >I know that the sphere doesn't embed in R^2, I wouldn't necessarily > >know hoe to prove it, though... > > >The invariance of dimension theorem says that R^m=R^n iff m=n, right? > > That's one very special case of the invariance of domain theorem. > > >I am not sure how to use it to prove what i want..could you give me > >more details, please? > > First you need to find out what the theorem actually says. I could > just tell you, but since you evidently have access to the internet > it would probably be more instructive for you to simply look it > up. On wikipedia for example. Thank you. I was looking up invariance of dimension; only now I looked up invariance of domain. Also, I found: "When the domain manifold is compact, the notion of a smooth embedding is equivalent to that of an injective immersion." and "An immersion is a local embedding (i.e. for any point x\in M there is a neighborhood x\in U\subset M such that f:U\to N is an embedding.)", so using these and the invariance of domain, we get that for every x in M we have a nbhd which is homeomorphic to its image under this embedding, right? How does this get a contradiction, though?
From: José Carlos Santos on 11 Jan 2010 04:41 On 10-01-2010 23:23, Maximilian Rogers wrote: >>>>> How can I show that a compact n- manifold does not embed in R^n? >> >>>> Does the unit sphere S^2, embed into R^2? >> >>> I know that the sphere doesn't embed in R^2, I wouldn't necessarily >>> know hoe to prove it, though... >> >>> The invariance of dimension theorem says that R^m=R^n iff m=n, right? >> >> That's one very special case of the invariance of domain theorem. >> >>> I am not sure how to use it to prove what i want..could you give me >>> more details, please? >> >> First you need to find out what the theorem actually says. I could >> just tell you, but since you evidently have access to the internet >> it would probably be more instructive for you to simply look it >> up. On wikipedia for example. > > Thank you. I was looking up invariance of dimension; only now I looked > up invariance of domain. Then you should be able now to prove that the image of an embedding of a n-dimensional manifold M into R^n is an open subset of R^n. Best regards, Jose Carlos Santos
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