In the previous paper, 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 [3] for immersed surfaces in the affine
$3$-space.