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

Our principal goal is to study the prescribed curvature problem in a manifold with density. In particular, we consider the Euclidean 3-space ${\mathbb{R}}^3$ with a positive density function $e^{\varphi }$ , where $\varphi =-x^2-y^2$ , $(x,y,z)\in {\mathbb{R}}^3$ and construct all the helicoidal surfaces in the space by solving the second-order non-linear ordinary differential equation with the weighted Gaussian curvature and the mean curvature functions. As a result, we give a classification of weighted minimal helicoidal surfaces as well as examples of helicoidal surfaces with some weighted Gaussian curvature and mean curvature functions in the space.

In this paper, we study submanifolds in a Euclidean space with a generalized 1-type Gauss map. The Gauss map, G, of a submanifold in the n-dimensional Euclidean space, ${\mathbb{E}}^n$, is said to be of generalized 1-type if, for the Laplace operator, $\Delta$, on the submanifold, it satisfies $\Delta G=fG+gC$, where C is a constant vector and f and g are some functions. The notion of a generalized 1-type Gauss map is a generalization of both a 1-type Gauss map and a pointwise 1-type Gauss map. With the new definition, first of all, we classify conical surfaces with a generalized 1-type Gauss map in ${\mathbb{E}}^3$. Second, we show that the Gauss map of any cylindrical surface in ${\mathbb{E}}^3$ is of the generalized 1-type. Third, we prove that there are no tangent developable surfaces with generalized 1-type Gauss maps in ${\mathbb{E}}^3$, except planes. Finally, we show that cylindrical hypersurfaces in ${\mathbb{E}}^{n+2}$ always have generalized 1-type Gauss maps.