On Projective Umbilics: a Geometric Invariant and an Index

Ricardo Uribe-Vargas (1)

1 CdF (institution) - Collège de France

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.

Domaines: Mathématiques [math]; Géométrie différentielle [math.DG]

Ricardo Uribe-Vargas. On Projective Umbilics: a Geometric Invariant and an Index. Journal of Singularities, 2018, 17, pp.81 - 90. ⟨10.5427/jsing.2018.17e⟩. ⟨hal-01741660⟩