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.



Local vs. infinitesimal rigidity
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](http://library.msri.org/books/Book54/files/02fisher.pdf), where Weil's theorem is Theorem 3.2.


See also Section 5 of [this paper](https://projecteuclid.org/journals/experimental-mathematics/volume-15/issue-3/Computing-Varieties-of-Representations-of-Hyperbolic-3-Manifolds-into-SLfR/em/1175789760.full).


