Lines of affine principal curvatures of surfaces in 3-space (Q2301584)

The authors study the principal lines in the context of affine differential geometry of surfaces immersed in 3-space. They consider the binary differential equation of the affine curvature lines and obtain topological models of these curves near the affine umbilic points (elliptic and hyperbolic). They also describe the local behavior of affine principal lines near affine umbilic points (elliptic and hyperbolic), points with double eigenvalues of the affine shape operator (but not umbilics) and parabolic points. As a result, they prove that for a generic surface, at regular parabolic points, the parabolic curve is an affine curvature line.