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Suppose that $M$ is an embedded sub-manifold of $D$-dimensional Euclidean space $E^D$, with embedding $\phi:M \hookrightarrow E^D$; the embedding is not explicitly known.

And suppose that I know the induced Riemannian metric $g$ on $M$, which depends by construction on $\phi$ (which so happens that I don't explicitly know).

Can I express $\phi$ in terms of $g$ (closed form or as a PDE maybe?), supposing that I want $\phi$ to be the embedding into $M$ which has minimal normal curvature and contains a predetermined (fixed) set of points.

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    $\begingroup$ First note that you can of course at best expect to get $\phi$ up to Euclidean isometry. Even so, the answer is negative in general, as mentioned by Ben McKay (and his comment is probably the best answer to that). Now, I think that there exist some rigidity result in this direction, when $(M,g)$ satisfies some hypotheses. If you are interested in a particular situation, it might help to give more information. $\endgroup$
    – Benoît Kloeckner
    Jul 3, 2016 at 12:46
  • $\begingroup$ In the case where my surface cannot deform would this be possible? $\endgroup$
    – AIM_BLB
    Jul 3, 2016 at 14:02
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    $\begingroup$ In what sense do you want to "minimize normal curvature"? $\endgroup$
    – Willie Wong
    Jul 3, 2016 at 17:41
  • $\begingroup$ it minimizes: $\int_M |k_n|^2 dm$, where m is the Remannian volume measure $\endgroup$
    – AIM_BLB
    Jul 4, 2016 at 4:14
  • $\begingroup$ The round sphere in Euclidean 3-space is rigid, but a punctured sphere is not, so you have to expect some global aspect to the problem. It is not enough to use local methods. $\endgroup$
    – Ben McKay
    Jul 4, 2016 at 8:20

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If the dimension of $M$ is greater than $2$, then the rigidity of $M$ can be due to the fact that the Gauss equations (giving the Riemannian curvature in terms of the second fundamental form) have a unique solution for the second fundamental form. In that case, it is, in principle, possible to reconstruct the embedding by first solving for the (unique) second fundamental form satisfying the Gauss equations. The embedding can then be "reconstructed" by integrating the system of partial equations that give the Riemannian metric and second fundamental form in terms of the embedding.

If $M$ is a closed $2$-dimensional convex surface in $\mathbb{R}^3$, then it is also known to be rigid. However, this is proved using an integral identity. In this case, it is not at all clear how to reconstruct the embedding. There are possible strategies, but they all seem impossible to carry out using known techniques.

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  • $\begingroup$ That is very interesting and can provide a very good start from me... would you happen to have a link to some references to any of these (it would provide a great starting point to me); also what do you mean by integrating the system of PDEs in terms of the embedding, thank you! :) $\endgroup$
    – AIM_BLB
    Jul 4, 2016 at 4:02
  • $\begingroup$ For the case when the dimension of $M$ is greater than $2$, see publications.ias.edu/sites/default/files/gausseq.pdf $\endgroup$
    – Deane Yang
    Jul 4, 2016 at 16:44
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    $\begingroup$ Here, when I say a system of PDE's can be integrated, I mean that it can be solved using the Frobenius theorem, which means that the system can be reduced to solving a set of ODE's. $\endgroup$
    – Deane Yang
    Jul 4, 2016 at 16:46
  • $\begingroup$ Perfect! Thanks for these notes this is very useful! :) $\endgroup$
    – AIM_BLB
    Jul 4, 2016 at 21:34
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No. Think about the helicoid, or other surfaces which deform.

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    $\begingroup$ Take a piece of paper in your hand, and hold it from one corner, and then turn your hand around. $\endgroup$
    – Ben McKay
    Jul 3, 2016 at 10:32
  • $\begingroup$ Thanks for the answer but I should have added in the condition that my surface cannot deform as an assumption $\endgroup$
    – AIM_BLB
    Jul 3, 2016 at 14:00

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