  In the previous paper (arXiv:0804.0701), the authors gave criteria for
A_{k+1}-type singularities on wave fronts. Using them, we show in this paper
that there is a duality between singular points and inflection points on wave
fronts in the projective space. As an application, we show that the algebraic
sum of 2-inflection points (i.e. godron points) on an immersed surface in the
real projective space is equal to the Euler number of M_-. Here M^2 is a
compact orientable 2-manifold, and M_-$is the open subset of M^2 where the
Hessian of f takes negative values. This is a generalization of Bleecker and
Wilson's formula for immersed surfaces in the affine 3-space.
