# Some rigidity stuff

While looking for results on rigidity, I stumbled upon the following paper by Fisher:

David Fisher, First cohomology and local rigidity of group actions, Ann. Math. (to appear ? a preprint is available since 2009)

The main result of this paper is:

Thm: Let \Gamma be a finitedly presented group, (M,g) a compact Riemannian manifold, and \pi: \Gamma \to Isom(M,g) \subset Diff^\infty(M) a homomorphism. If H^1′\Gamma, Vect^\infty(M)) = 0, then \pi is locally rigid as a homomorphism into Diff^\infty(M)

This paper also uses the Nash-Moser method (Hamilton’s result) to prove the above theorem. The author’s proof the tameness condition (in order to use Nash-Moser) is based on the fact that Isom(M,g) is compact, and is similar to Conn’s method (in his paper on linearization of Poisson structure)

Compare to our rigidity result obtained with Eva Miranda and Philippe Monnier: our rigidity result is for Hamiltonian actions on Poisson manifolds. The tameness condition is also based on Conn’s proof. We also use Nash-Moser. But we can’t use Hamilton’s theorems directly. Instead, we had to develop Nash-Moser a bit further into a local normal form theorem of Nash-Moser type.

We believe that there are more smooth rigidity results that one can show using our Nash-Moser normal form theorem :D.

Another rigidity paper which I found by chance is: Spatzier, An invitation to rigidity theory (in a volume dedicated to Anatoly Katok ?). This paper is a survey of structure rigidity in various contexts (Anosov flows, quasi-symmetries, symmetric spaces, curvature pinching, …) The paper contains no proofs, so it’s hard to see what methods are used.