Complete classification of surfaces with a canonical principal direction in the Euclidean space \(\mathbb{E}^{3}\) (Q539194)

A constant angle surface is defined as a surface whose unit normal forms a constant angle with a fixed direction \(\vec{k}\). The projection of its fixed direction \(\vec{k}\) on the tangent plane (denoted by \(U\)) is a principal direction with vanishing corresponding principal curvature. The study of constant angle surfaces in \(\mathbb{R}^3\) can be generalized for surfaces whose angle function is no longer constant, provided that certain properties are preserved. For example, when \(U\) is a principal direction with corresponding principal curvature different from \(0\), these surfaces are called surfaces with a canonical principal direction. In the paper, the authors classify them, construct some examples and prove that the only minimal surface with a canonical principal direction in the Euclidean space \(\mathbb{R}^3\) is the catenoid.