The horizontal Hessian matrix of the distance function in the Heisenberg group.

In a joint work with Nicola Arcozzi of the University of Bologna, we studied, in the Heisenberg group, a notion of metric normal, that is the geodesic gamma leaving a non characteristic point P on a smooth surface S such that locally every point Q that belongs to gamma satisfies d(Q; P) = d(Q; S). In particular we gave sufficient conditions about the existence of such metric normal for smooth surfaces far from characteristic points. We proved that there exist smooth surfaces S such that the horizontal Hessian matrix of the distance function from S are not bounded. Moreover starting from the horizontal Hessian matrix of the distance function from S, evaluated in the non characteristic points of the surface, we prove that the trace of such matrix coincide with the natural notion of curvaure of the surface S given in the Heisenberg group. (Received September 07, 2004)