On the patterns of principal curvature lines around a curve of umbilic points

Abstract

In this paper is studied the behavior of principal curvature lines near a curve of umbilic points of a smooth surface.

Surfaces around closed principal curvature lines, an inverse problem

“…The generic behavior of principal curvature lines near a regular curve of umbilics was studied in [3].…”

“…Proof. Consider the parametric surface defined by equation (3). By Proposition 1, c is a principal cycle when θ ′ (s) = −τ (s), θ(0) = θ 0 , L 0 τ (s)ds = 2mπ.…”

Lines of Curvature on Surfaces, Historical Comments and Recent Developments

“…The lines S i , are called separatrices of F 2 (α) at p. Similarly, for F 1 (α). In this subsection the results of [34] will be outlined. The interest on the structure of principal lines in a neighborhood of a continuum of umbilic points, forming a curve, in an analytic surface goes back to the lecture of Carathéodory [5].…”