Characteristic Points, Fundamental Cubic Form and Euler Characteristic of Projective Surfaces

Maxim Kazarian (1, 2) , Ricardo Uribe-Vargas (3, 4)

1 HSE - Vysšaja škola èkonomiki = National Research University Higher School of Economics [Moscow]
2 Skoltech - Skolkovo Institute of Science and Technology [Moscow]
3 IMB - Institut de Mathématiques de Bourgogne [Dijon]
4 LaSol - Laboratoire Solomon Lefschetz

We define local indices for projective umbilics and godrons (also called cusps of Gauss) on generic smooth surfaces in projective 3-space. By means of these indices, we provide formulas that relate the algebraic numbers of those characteristic points on a surface (and on domains of the surface) with the Euler characteristic of that surface (resp. of those domains). These relations determine the possible coexis-tences of projective umbilics and godrons on the surface. Our study is based on a "fundamental cubic form" for which we provide a simple closed expression.

Mots clés: cusp of Gauss; front; surface; projective umbilic; singularity; flecnodal curve; parabolic curve; index; Differential geometry; godron; quadratic point

Domaines: Géométrie différentielle [math.DG]; Topologie géométrique [math.GT]

Maxim Kazarian, Ricardo Uribe-Vargas. Characteristic Points, Fundamental Cubic Form and Euler Characteristic of Projective Surfaces. Moscow Mathematical Journal, 2020, 20 (3), pp.511-530. ⟨10.17323/1609-4514-2020-20-3-511-530⟩. ⟨hal-02568225v2⟩