  The motion of a block slipping on a surface is a well studied problem for
flat and circular surfaces, but the necessary conditions for the block to leave
(or not) the surface deserve a detailed treatment.
  In this article, using basic differential geometry, we generalize this
problem to an arbitrary surface, including the effects of friction, providing a
general expression to determine under which conditions the particle leaves the
surface. An explicit integral form for the velocity is given, which is
analytically integrable for some cases, and we find general criteria to
determine the critical velocity at which the particle immediately leaves the
surface. Five curves, a circle, ellipse, parabola, catenary and cycloid, are
analyzed in detail. Several interesting features appear, for instance, in the
absense of friction, a particle moving on a catenary leaves the surface just
before touching the floor, and in the case of the parabola, the particle never
leaves the surface, regardless of the friction. A phase diagram that separates
the conditions that lead to a particle stopping in the surface to those that
lead to a particle departuring from the surface is also discussed.
