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In this paper we discuss the behavior of the curvature lines of a transversal eq\"uiaffine vector field along a surface in $3$-space at isolated umbilical points., Comment: 12 pages. This article was originally part of earlier versions of arXiv: 1811.07331 arXiv admin note: text overlap with arXiv:1811.07331

There is a natural duality between line congruences in $\mathbb{R}^3$ and surfaces in $\mathbb{R}^4$ that sends principal lines into asymptotic lines. The same correspondence takes the discriminant curve of a line congruence into the parabolic curve of the dual surface. Moreover, it takes the ridge curves to the flat ridge curves, while the subparabolic curves of a line congruence are taken to certain curves on the surface that we call flat subparabolic curves. In this paper, we discuss these relations and describe the generic behavior of the subparabolic curves at the discriminant curve of the line congruence, or equivalently, the parabolic curve of the dual surface. We also discuss Loewner's conjectures under the duality viewpoint., Comment: 17 pages, 2 figures