Some remarks on Legendrian rectifiable currents

A rectifiable current of dimension n - 1 in the sphere bundle SRn similar or equal to R-n x Sn-1 for euclidean space is Legendrian if it annihilates the contact 1-form alpha (i.e. T(alpha boolean AND phi) = 0 for all forms phi of degree n - 2). Such a current may be naturally associated to any convex set or to any singular real analytic variety, and induces the curvature measures of such a set. We prove that the projection to R-n of a carrier of a general such T is C-2-rectifiable in the sense of Anzellotti-Serapioni. We deduce that the boundary of a set with positive reach, as well as its singular skeleta, are C-2-rectifiable. In case partial derivative T = 0 we prove also that the curvature measures associated to T satisfy the analogues of the classical variational formulas for curvature integrals. It follows that such formulas are valid for the curvature measures of subsets of space forms.