On Projective Umbilics: a Geometric Invariant and an Index

Journal of Singularities
volume 17 (2018), 81-90

Received: 7 December 2017. Received in revised form: 6 March 2018.

DOI: 10.5427/jsing.2018.17e

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Abstract:

We define a geometric invariant and an index (+1 or -1) for projective umbilics of smooth surfaces. We prove that the sum of the indices of the projective umbilics inside a connected component H of the hyperbolic domain remains constant in any 1-parameter family of surfaces if the topological type of H does not change. We prove the same statement for any connected component E of the elliptic domain. We give formulas for the invariant and for the index which do not depend on any normal form.

Keywords:

Differential geometry, surface, singularity, parabolic curve, flecnodal curve, projective umbilic, invariant, index, cross-ratio, quadratic point

Mathematical Subject Classification (2010):

53A20, 53A55, 53D10, 57R45, 58K05

Author(s) information:

Ricardo Uribe-Vargas
Institut de Mathématiques de Bourgogne
UMR 5584 CNRS
Univ. Bourgogne Franche-Comté
email: r.uribe-vargas@u-bourgogne.fr