On the multiplicity of umbilic points

TL;DR

This paper introduces a new invariant called multiplicity for umbilic points on surfaces in Euclidean or Minkowski 3-space, which measures the maximum number of stable umbilic points resulting from surface deformations.

Contribution

It defines and analyzes the properties of the multiplicity invariant for umbilic points, providing a novel way to understand their stability and splitting behavior.

Findings

Defines the multiplicity invariant for umbilic points.

Establishes properties of the multiplicity invariant.

Provides insights into the stability and splitting of umbilic points.

Abstract

We introduce an invariant of umbilic points on surfaces in the Euclidean or Minkowski 3-space that counts the maximum number of stable umbilic points they can split up under deformations of the surfaces. We call that number the multiplicity of the umbilic point and establish its properties.