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accepted | How useful/pervasive are differential forms in surface theory? |
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How useful/pervasive are differential forms in surface theory?
Thanks Richard. Do you have a pointer to a good proof of Gauss-Bonnet using forms? I typically have the students prove the discrete analog (via angle defect), but have so far not found a nice, simple proof of the smooth version that uses forms. |
Mar
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awarded | Nice Question |
Mar
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How useful/pervasive are differential forms in surface theory?
Thank you Deane; thank you Robert. These answers are in line with what was my fuzzy view of the culture, and it's very nice to have these concrete reference points and examples. (I am also very tempted to use this proof of the theorema egregium in my class!) Thanks again. |
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How useful/pervasive are differential forms in surface theory?
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asked | How useful/pervasive are differential forms in surface theory? |
Feb
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accepted | How hard is it to determine if a weighted graph can be isometrically embedded in R^3? |
Feb
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How hard is it to determine if a weighted graph can be isometrically embedded in R^3?
Great. (And thanks for some very interesting pointers.) So to summarize crudely: for general graphs it is hard but there are relaxations; for convex triangulated surfaces it can be solved efficiently for approximation but is hard or impossible to do exactly. No statement on the difficulty of finding approximate solutions for nonconvex surfaces. |
Feb
9 |
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How hard is it to determine if a weighted graph can be isometrically embedded in R^3?
(E.g., in Figure 1 the extrinsic length of the arc will match the intrinsic length of the corresponding edge.) |
Feb
9 |
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How hard is it to determine if a weighted graph can be isometrically embedded in R^3?
Nice paper! But here the distance they consider is the geodesic rather than Euclidean distance induced by the embedding, no? |
Feb
9 |
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How hard is it to determine if a weighted graph can be isometrically embedded in R^3?
P.S. Yes, I mean a triangulated surface. |
Feb
9 |
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How hard is it to determine if a weighted graph can be isometrically embedded in R^3?
Thanks Joseph. Yes, these kinds of finite isometries do come to mind, though I don't yet have much intuition for whether they make it hard to find just one isometric embedding. From an optimization point of view: if the surface is infinitesimally rigid then any objective function with zeros only at isometric embeddings must be nonconvex (the zeros are isolated). But to answer the existence question, one may need only ski downhill to the bottom of a single valley... |
Feb
8 |
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How hard is it to determine if a weighted graph can be isometrically embedded in R^3?
Excellent. Thanks. So the only question remaining is the second one: do things get easier for simplicial graphs? |