Rigidity properties for some isometric extensions of partially hyperbolic actions on the torus

This paper studies local rigidity for some isometric toral extensions of partially hyperbolic $\mathbb{Z}^k$ ($k\geqslant 2$) actions on the torus. We prove a $C^\infty$ local rigidity result for such actions, provided that the smooth perturbations of the actions satisfy the intersection property. We also give a local rigidity result within a class of volume preserving actions. Our method mainly uses a generalization of the KAM iterative scheme.