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Say you have a 2-sphere endowed with a metric. You embed this into $\mathbb{R}^3$. The intrinsic metric won't match the actual distances between neighbouring points in $\mathbb{R}^3$.

If you calculated the total error of the mismatch, you might be able slightly deform the sphere to get a better embedding. I guess the error function would look something like:

$$ E[X] = \int_S \sum_{ab}|g_{ab}(\sigma) - \partial_a X^\mu(\sigma)\partial_b X^\mu(\sigma)|^2 d\sigma^2 $$

Then perhaps you could deform the sphere and by gradient descent try to minimise the error to get a better embedding. Would you always get to a solution? Or would it get stuck in a local minimum?

Another way to look at it would be, imagine taking a closed surface made of some elastic material and stretched it over a balloon. Then you popped the balloon, would the surface go back to it's original shape? Would you need some sort of drag term to stop it just vibrating forever?

(There is a theorem that every closed 2D manifold with analytic metric can be analytically embedded into $\mathbb{R}^3$. But this is an existence theorem it doesn't tell you how to do it.)

Edit: might have to assume it has positive curvature everywhere. (although this is not as interesting!) Or you may assume that at least one solution is possible.

(Weyl-Lewy Nirenberg-Pogorelov) Any analytic (smooth ) positive cur- vature metric defined on S2 always admits an analytic ( a smooth ) isometric em- bedding in R3.

source https://arxiv.org/pdf/math/0304391.pdf

Edit: Also, is this at all related to Ricci flow?

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  • $\begingroup$ Good point. It might just apply to genus 0 or those without negative curvature. Let me check. $\endgroup$
    – zooby
    Jan 10 '18 at 17:44
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More an extended comment than an answer: I think a challenge you encounter is that the gradient descent procedure is not guaranteed to keep the embedding function within the space of smooth (or even $C^2$) functions: it could try to add creases or cone singularities to the manifold. In fact I recently ran a related simulation, where I was trying to recover a hemisphere by prescribing the conformal metric given by stereographic projection to the flat disk. The descent procedure halted here (which, at least, matches physical intuition about rubber surfaces getting "stuck" in metastable states):

enter image description here

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    $\begingroup$ That's very interesting. I should like to have a go at making a simulation like this! $\endgroup$
    – zooby
    Jan 10 '18 at 18:05

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