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

Hello expensive customer to our community We will proffer you an answer to this query mg.metric geometry – Infinitesimal rigidity vs native rigidity of isometrically immersed riemannian manifolds ,and the respond will breathe typical by documented data sources, We welcome you and proffer you fresh questions and solutions, Many customer are questioning concerning the respond to this query.

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.

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.