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Complete classification of surfaces with a canonical principal direction in the Euclidean space $$\mathbb{E}^{3}$$. (English) Zbl 1222.53009
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.

MSC:
 53A05 Surfaces in Euclidean and related spaces
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References:
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