Timeline for Isometric but differently shaped closed surfaces in $\mathbb{R}^3$
Current License: CC BY-SA 3.0
20 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Feb 28, 2013 at 22:16 | comment | added | Gerald Edgar | When you are making your band out of a strip of paper, go ahead and put any number of twists in it. | |
| Feb 28, 2013 at 22:07 | answer | added | Misha | timeline score: 7 | |
| Feb 28, 2013 at 21:56 | comment | added | Misha | Hans: Then you should make further modifications of your question removing "I wonder if" and stressing that by "examples" you want "examples with explicit isometric embedding in $R^3$." | |
| Feb 28, 2013 at 21:36 | answer | added | Robert Young | timeline score: 6 | |
| Feb 28, 2013 at 21:34 | history | edited | Hans-Peter Stricker | CC BY-SA 3.0 |
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| Feb 28, 2013 at 21:33 | comment | added | Hans-Peter Stricker | @Misha: That's exactly what I meant. The only thing I did not make clear enough is, that I want to (and need to) see pictures! (I added an according remark.) | |
| Feb 28, 2013 at 21:21 | comment | added | Misha | Hans: You have several questions here. One of them is (to the best of my understanding): Can one have a surface $S$ and two smooth maps $f_i: S\to R^3$ so that the pull-back Riemannian metrics have identical curvature functions but different metrics. To construct such, take positively curved examples in MO 100281 and apply Pogorelov embedding theorem. Of course, maybe you meant something else, but then you should clarify your question. | |
| Feb 28, 2013 at 21:12 | comment | added | Hans-Peter Stricker | @Misha: I am aware of the fact that curvature does not determine the metric, and I followed the (very) discussion you point me to - thanks for that. But I'd like to see it! (I find it hard to make immediate sense of your last remark: what do you want to tell me with respect to my question?) | |
| Feb 28, 2013 at 21:05 | comment | added | Misha | The fact that curvature does not determine the metric (for surfaces) is discussed in great detail here: mathoverflow.net/questions/100281/… If you are interested in surfaces in $R^3$, note that every positively curved surface embeds isometrically as a convex surface in $R^3$. | |
| Feb 28, 2013 at 20:59 | answer | added | Lee Mosher | timeline score: 2 | |
| Feb 28, 2013 at 20:11 | comment | added | Hans-Peter Stricker | I assume you want to "show" me - "just imagine..." - two different embeddings of the Moebius band into $\mathbb{R}^3$. One of them - the first (standard) one? - probably is this: mathworld.wolfram.com/images/eps-gif/MobiusStrip_1000.gif. But I have no idea, how the other one looks like. | |
| Feb 28, 2013 at 19:47 | comment | added | Ryan Budney | Just imagine making a Moebius band out of a thin strip of paper. There's two standard bends you can put in it -- the kind of bend you have in the standard embedding of the cylinder $S^1 \times [0,1]$ in $\mathbb R^3$, and then there's the planar embedding of the cylinder, in polar coordinates $1 \leq r \leq 2$. | |
| Feb 28, 2013 at 19:28 | comment | added | Hans-Peter Stricker | Concerning curvature: a) I have difficulties to imagine two non-isometric zero-curvature Moebius bands embedded in $\mathbb{R}^3$. b) Even if I managed: it would be a surface with a boundary. | |
| Feb 28, 2013 at 19:26 | comment | added | Hans-Peter Stricker | Concerning the metric: I hoped not to have to be too specific: math.stackexchange.com/questions/315710/… (Myers-Steenrod theorem). | |
| Feb 28, 2013 at 18:25 | comment | added | Ryan Budney | Also, what do you mean when you say the surfaces have the "same" Gaussian curvature, just a diffeomorphism that carries the Gauss curvature? If so, there's an immense variety of such surfaces -- the zero-curvature embeddings of the Moebius band gets you started, but there are similar things for hyperbolic surfaces. | |
| Feb 28, 2013 at 18:21 | comment | added | Ryan Budney | Which metric are you talking about? | |
| Feb 28, 2013 at 17:52 | history | edited | Hans-Peter Stricker | CC BY-SA 3.0 |
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| Feb 28, 2013 at 17:25 | history | asked | Hans-Peter Stricker | CC BY-SA 3.0 |
