On Projective Umbilics: a Geometric Invariant and an Index

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.

13 pages, 7 figures

References in corpus (1)

• Characteristic Points, Fundamental Cubic Form and Euler Characteristic of Projective Surfaces