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Can someone please explain the difference between local rigidity and infinitesimal rigidity? Does either version of rigidity imply the other?

In particular, I'm thinking about Weil's rigidity theorem for hyperbolic metrics on manifolds of dimension $\geq 3$. I've seen it referred to as both local and infinitesimal, which further adds to my confusion about the distinction.

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+1 This is a good question. If you find an answer off-MO, please repeat it here? – Sam Nead Aug 5 '10 at 19:15
up vote 12 down vote accepted

Infinitesimal rigidity implies local rigidity, but not conversely. Local rigidity means a representation has no deformations, whereas infinitesimal rigidity means the natural tangent space to the character variety is 0-dimensional (this tangent space is a certain cohomology group with twisted coefficients). Weil proved both in the context you mention. See e.d. David Fisher's survey paper, where Weil's theorem is Theorem 3.2.

See also Section 5 of this paper.

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Thanks, Nathan! – Dave Futer Aug 5 '10 at 21:28

Local rigidity means that the structure in question is an isolated point in its deformation space (which is typically an algebraic set). Infinitesimal rigidity means that there are no first-order deformations of the structure in question. A first-order deformation is a nonzero element of a certain cohomology group.

Because you can take the derivative of a path of structures and get a first-order deformation, infinitesimal rigidity implies local rigidity.

Because a first-order deformation may or may not correspond to an actual path (due to higher-order obstructions), local rigidity does NOT necessarily imply infinitesimal rigidity.

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It appears I can only "accept" one answer, but I really liked this explanation too. – Dave Futer Aug 5 '10 at 21:27

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