In this paper, we introduce two notions on a surface in a contact
manifold. The first one is called degree of transversality (DOT) which
measures the transversality between the tangent spaces of a surface
and the contact planes. The second quantity, called curvature of
transversality (COT), is designed to give a comparison principle for
DOT along characteristic curves under bounds on COT. In particular,
this gives estimates on lengths of characteristic curves assuming COT
is bounded below by a positive constant.
We show that surfaces with constant COT exist and we classify all
graphs in the Heisenberg group with vanishing COT. This is
accomplished by showing that the equation for graphs with zero COT can
be decomposed into two first order PDEs, one of which is the backward
invisicid Burgers' equation. Finally we show that the p-minimal graph
equation in the Heisenberg group also has such a
decomposition. Moreover, we can use this decomposition to write down
an explicit formula of a solution near a regular point.