Local vs. infinitesimal rigidity - MathOverflow

Local vs. infinitesimal rigidity

2010-08-05T15:13:25Z

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.

Answer by Nathan Dunfield for Local vs. infinitesimal rigidity

2010-08-05T20:37:50Z

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.

Answer by David Dumas for Local vs. infinitesimal rigidity

2010-08-05T20:43:21Z

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.