On Projective Umbilics: a Geometric Invariant and an Index

TL;DR

This paper introduces a new geometric invariant and index for projective umbilics on smooth surfaces, proving their topological invariance within hyperbolic and elliptic domains during continuous surface deformations.

Contribution

It defines a novel invariant and index for projective umbilics and proves their invariance under certain topological conditions without relying on normal forms.

Findings

The sum of indices remains constant in hyperbolic domains during surface deformations.

The sum of indices remains constant in elliptic domains during surface deformations.

Formulas for the invariant and index are provided independent of normal forms.

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.