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mg.metric geometry – Infinitesimal rigidity vs native rigidity of isometrically immersed riemannian manifolds

I used to be studying the good survey on rigidity, specializing in tensegrities by Connelly, and I’d love to know the standing and suggestions a few query he asks:

**Theorem 3.1** within the above, says that for cable/swagger/barrier finite frameworks, native flexibility (i.e. the actuality of non-trivial steady deformations which respect the tensegrity constraints) is equal to infinitesimal flexibility (particularly, the situations one obtains on the first jet point from the above: explicitly, there live $p_i’$ “tangent vectors” at configuration factors $p_i$, satisfying unaffected constraints reminiscent of $(p_i-p_j)cdot(p_i’-p_j’)le 0$ if ${p_i,p_j}$ is a cable, and analogues for bars and struts).

The proof of 1 implication is trifling, the opposite implication follows by algebraic regularization.

**Connelly asks for a steady analogue of this outcome.**

One may give an **interpretation of this query** by asking: is an isometric immersion of a Riemannian manifold in Euclidean house *domestically elastic* (within the sense that there’s a path of isometric immersions that passes by it, not coming from a path of isometries of ambient house) if and solely whether it is *infinitesimally elastic* (=situation obtained by taking the first jet, or tangent vectorfield, model of the above)?

I’m inquisitive if the above interpretations appears legit, and if some solutions are identified, particularly within the setting of immersions of $C^{1,alpha}$-regularity for $alpha$ diminutive (or zero). In that illustration, one might be conjectural to take feeble derivatives within the definition of infinitesimal rigidity. Or alternatively, give one other interpretation of the query.

The accent on regularity is motivated by the illustrious rigidity outcomes by Cohn-Vossen / Pogorelov / Nirenberg / Borisov, which labor at greater regularity. If all isometric immersions are inflexible, then the query turns into much less fascinating. Flexibility has been proved at scowl $alpha$ by De Lellis – Inauen – Szekelihidi (amongst others) following a path began by Nash, and this makes the query maybe extra fascinating in that setting. For a greater dialogue on this behold this survey.

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