Rigidity of group actions, cotangent lifts and integrable systems

Abstract

In this master thesis we generalize a theorem by Palais on the rigidity of compact group actions to cotangent lifts. We use this result to prove rigidity for integrable systems on symplectic manifolds including systems with degenerate singularities which are invariant under a torus action. We also prove the $b$-symplectic analogue of the rigidity results. We illustrate the three basic types of singularities of integrable systems through three models from classical mechanics and we give them as cotangent lifts. Finally we review the focus-focus singularity and the saddle-focus singularity.