Translation surfaces in the Galilean 3-space G3 have two types according to the isotropic and non-isotropic plane curves. In this paper, we study a translation surface in G3 with a log-linear density and classify such a surface with vanishing weighted mean curvature.

We study rotation surfaces in the three-dimensional pseudo-Galilean space G₃¹ such that the Gauss map G satisfies the condition L₁G = f(G + C) for a smooth function f and a constant vector C, where L₁ is the Cheng-Yau operator.

We study quadric surfaces in Euclidean 3-space with non-degenerate second fundamental form, and classify them in terms of the Gaussian curvature, the mean curvature, the second Gaussian curvature and the second mean curvature.

The family of cones is one of typical models of non-cylindrical ruled surfaces. Among them, the circular cones are unique in the sense that their Gauss map satisfies a partial differential equation similar, though not identical, to one characterizing the so-called 1-type submanifolds. Specifically, for the Gauss map G of a circular cone, one has ΔG = f(G+C), where Δ is the Laplacian operator, f is a non-zero function and C is a constant vector. We prove that circular cones are characterized by being...

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