On Rigidity of Generalized Conformal Structures

The classical Liouville Theorem on conformal transformations determines local conformal transformations on the Euclidean space of dimension $\ge 3$. Its natural adaptation to the general framework of Riemannian structures is the 2-rigidity of conformal transformations, that is such a transformation is fully determined by its 2-jet at any point. We prove here a similar rigidity for generalized conformal structures defined by giving a one parameter family of metrics (instead of scalar multiples of a given one) on each tangent space.