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I was reading the nice survey on rigidity, focusing on tensegrities by Connelly, and I'd like to know the status and feedback about a question he asks:

A theorem by Gluck and this work of Connelly, about cable/strut/bar finite frameworks, relate notions such as the following:

  1. local flexibility (i.e. the existence of non-trivial continuous deformations which respect the tensegrity constraints)
  2. infinitesimal flexibility (namely, the conditions one obtains at the first jet level from the above: explicitly, there exist $p_i'$ "tangent vectors" at configuration points $p_i$, satisfying natural constraints such as $(p_i-p_j)\cdot(p_i'-p_j')\le 0$ if $\{p_i,p_j\}$ is a cable, and analogues for bars and struts).

Gluck shows that 1 implies 2 and that for generic frameworks, 1 is equivalent to 2. (As noted in the comments, in without a further genericity assumption, the viceversa is not true.)

Connelly asks for a continuous analogue of such result.

One can give an interpretation of this question by asking: is an isometric immersion of a Riemannian manifold in Euclidean space locally flexible (in the sense that there is a path of isometric immersions that passes through it, not coming from a path of isometries of ambient space) if and only if it is infinitesimally flexible (=condition obtained by taking the 1st jet, or tangent vectorfield, version of the above)?

I am curious if the above interpretations seems legit, and if some answers are known, especially in the setting of immersions of $C^{1,\alpha}$-regularity for $\alpha$ small (or zero). In that case, one is probably supposed to take weak derivatives in the definition of infinitesimal rigidity. Or alternatively, give another interpretation of the question.

The emphasis on regularity is motivated by the famous rigidity results by Cohn-Vossen / Pogorelov / Nirenberg / Borisov, which work at higher regularity. If all isometric immersions are rigid, then the question becomes less interesting. Flexibility has been proved at lower $\alpha$ by De Lellis - Inauen - Szekelihidi (amongst others) following a path started by Nash, and this makes the question perhaps more interesting in that setting. For a better discussion on this see this survey.

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    $\begingroup$ I believe that your question is wide open. I do, however, want to call attention to the work of Berger-Bryant-Griffiths (projecteuclid.org/euclid.dmj/1077303336), who prove rigidity results by studying when the Gauss, Codazzi, and Ricci equations have unique solutions pointwise solutions. In particular, the rigidity theorem they prove where curvature tensor uniquely determines the second fundamental form can be generalized to situations where the Riemannian metric is at least $C^2$ and the isometric embedding is assumed to be only $W^{2,2}$. $\endgroup$
    – Deane Yang
    Commented Jan 10, 2021 at 22:42
  • $\begingroup$ I did not read the cited paper but the claimed result that local rigidity for frameworks is equivalent to infinitesimal rigidity is simply false. What's true is that infinitesimal rigidity for frameworks implies local rigidity. $\endgroup$
    – Moishe Kohan
    Commented Jan 10, 2021 at 23:10
  • $\begingroup$ @MoisheKohan true, I was superficial in my relation of the above statement, I will correct it. $\endgroup$
    – Mircea
    Commented Jan 11, 2021 at 15:49
  • $\begingroup$ @DeaneYang thanks, probably this paper is a good place to start in the above. You think that the generalization to $W^{2,2}$ is doable by classical techniques? $\endgroup$
    – Mircea
    Commented Jan 11, 2021 at 23:29
  • $\begingroup$ Yes, it's a straightforward argument. $\endgroup$
    – Deane Yang
    Commented Jan 11, 2021 at 23:33

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