Session S39 - Differential Equations and Geometric Structures

Tuesday, July 13, 13:00 ~ 13:50 UTC-3

On the multiplicity of umbilic points

Farid Tari

University of São Paulo, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Umbilic on a regular surface $M$ in the Minkowski 3-space $\mathbb R^3_1$ are the singular points of the lines of principal curvature (these can be extended at points where the induced pseudo metric on $M$ is degenerate). They are the points where all the coefficients of the binary differential equation (BDE) of these lines vanish.

Umbilic points on a generic surface are stable, that is, they persist and with the same local configuration of the lines of principal curvature under small deformations of the surface. However, the BDE of these lines is not stable when deformed within the set of all BDEs.

An invariant of a BDE, called the multiplicity of the BDE, at its singular point was introduced in [Bruce-Tari, 1998] and counts the maximum number of well folded singularities that can appear in a local deformation the BDE. Here, we introduce an invariant of an analytic surface $M$ in the Minkowski 3-space at its umbilic points, and call it the multiplicity of the umbilic point (the concept is also valid for surfaces in the Euclidean 3-space). The multiplicity counts the maximum number of stable umbilic points that can appear under small deformations of the surface at a non-stable umbilic point. We establish its properties and compute it in various cases.

Joint work with Marco Antonio do Couto Fernandes (University of Sao Paulo, Brazil).

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