The duality between singular points and inflection points on wave fronts

Details The duality between singular points and inflection points on wave fronts The duality between singular points and inflection points on wave fronts

2009-02-04

arxiv.org/abs/0902.0649v6

Mathematics

Differential Geometry

12 pages, one figure; to appear in Osaka Journal of Mathematics

Scientific paper

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.

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