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.