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Mircea
Mircea
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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:

Theorem 3.1 in the above A theorem by Gluck and this work of Connelly, says that forabout cable/strut/bar finite frameworks, local flexibility (i.e. the existence of non-trivial continuous deformations which respectrelate notions such as the tensegrity constraints)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 infinitesimal flexibility2. (namely, the conditions one obtains at the first jet level fromAs noted in the above: explicitly, there exist $p_i'$ "tangent vectors" at configuration points $p_i$comments, satisfying natural constraints such as $(p_i-p_j)\cdot(p_i'-p_j')\le 0$ if $\{p_i,p_j\}$ isin without a cable, and analogues for bars and struts).

The proof of one implication is trivialfurther genericity assumption, the other implication follows by algebraic regularizationviceversa is not true.)

Connelly asks for a continuous analogue of thissuch 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.

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:

Theorem 3.1 in the above, says that for cable/strut/bar finite frameworks, local flexibility (i.e. the existence of non-trivial continuous deformations which respect the tensegrity constraints) is equivalent to 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).

The proof of one implication is trivial, the other implication follows by algebraic regularization.

Connelly asks for a continuous analogue of this 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.

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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Mircea
Mircea
  • 2k
  • 1
  • 14
  • 24

Infinitesimal rigidity vs local rigidity of isometrically immersed riemannian manifolds

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:

Theorem 3.1 in the above, says that for cable/strut/bar finite frameworks, local flexibility (i.e. the existence of non-trivial continuous deformations which respect the tensegrity constraints) is equivalent to 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).

The proof of one implication is trivial, the other implication follows by algebraic regularization.

Connelly asks for a continuous analogue of this 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.