Inserted: 31 mar 2009
We investigate certain second-oder differential properties of functions and forms of class $C^1$ at the points around which a suitable Legendrian condition is \lq\lq very densely verified\rq\rq. In particular we provide a generalization of the classical equality $d^2=0$ for differential forms and some results about second-order osculating properties of graphs. Particular emphasis is placed on the case when the condition is verified in a locally finite perimeter set. A conjecture about the $C^2$-rectifiability of the horizontal projection of a Legendrian rectifiable set is discussed.