Moscow Mathematical Journal

Volume 20, Issue 3, July–September 2020 pp. 511–530.

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

Authors: Maxim Kazarian (1) and Ricardo Uribe-Vargas (2)
Author institution:(1) National Research University Higher School of Economics, Moscow, Russia; and Skolkovo Institute of Science and Technology, Moscow, Russia
(2) Laboratory Solomon Lefschetz UMI2001 CNRS, Universidad Nacional Autonoma de México, México City; and
Institut de Mathématiques de Bourgogne, UMR 5584, CNRS, Université Bourgogne Franche-Comté, F-21000 Dijon, France

Summary:

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 coexistences of projective umbilics and godrons on the surface. Our study is based on a “fundamental cubic form”, for which we provide a simple expression.

2010 Math. Subj. Class. 53A20, 53A55, 53D10, 57R45, 58K05

Keywords: Differential geometry, surface, front, singularity, parabolic curve, flecnodal curve, index, projective umbilic, quadratic point, godron, cusp of Gauss.

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