MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
+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 13 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.

share|cite|improve this answer
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.

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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.