  We define a geometric invariant and an index (+1 or -1) for projective
umbilics of smooth surfaces. We prove that the sum of the indices of the
projective umbilics inside a connected component H of the hyperbolic domain
remains constant in any 1-parameter family of surfaces if the topological type
of H does not change. We prove the same statement for any connected component E
of the elliptic domain. We give formulas for the invariant and for the index
which do not depend on any normal form.
