Related papers: Constant Angle Surfaces in a warped product

We classify all surfaces with constant Gaussian curvature $K$ in Euclidean $3$-space that can be expressed as an implicit equation of type $f(x)+g(y)+h(z)=0$, where $f$, $g$ and $h$ are real functions of one variable. If $K=0$, we prove…

In this article we generalize the notion of constant angle surfaces in S^2 x R and H^2 x R to general Bianchi-Cartan-Vranceanu spaces, i.e. essentially to three-dimensional homogeneous spaces with a four-dimensional isometry group. We show…

We define hypersurfaces $f\colon M^n\to \mathbb{Q}_{c_1}^{k} \times \mathbb{Q}_{c_2}^{n-k+1}$ in class $\mathcal{A}$ of a product of two space forms as those that have flat normal bundle when regarded as submanifolds of the underlying flat…

We explore convex shapes $S$ in the Euclidean plane which have the following property: there is a circle $C$ such that the angle between the two tangents from any point of $C$ to $S$ is constant equal to $\alpha$. A dynamical formulation…

In this article we study the shape of a compact surface of constant mean curvature of Euclidean space whose boundary is contained in a round sphere. We consider the case that the boundary is prescribed or that the surface meets the sphere…

In the last two decades, much effort has been dedicated to studying curves and surfaces according to their angle with a given direction. However, most findings were obtained using a case-by-case approach, and it is often unclear what are…

In Euclidean space, we investigate surfaces whose mean curvature $H$ satisfies the equation $H=\alpha\langle N,\mathbf{x}\rangle+\lambda$, where $N$ is the Gauss map, $\mathbf{x}$ is the position vector and $\alpha$ and $\lambda$ are two…

Surfaces of constant negative curvature in Euclidean space can be described by either the sine-Gordon equation for the angle between asymptotic directions, or a Monge-Ampere equation for the graph of the surface. We present the explicit…

A number of results for C$^2$-smooth surfaces of constant width in Euclidean 3-space ${\mathbb{E}}^3$ are obtained. In particular, an integral inequality for constant width surfaces is established. This is used to prove that the ratio of…

We give a complete description of all hypersurfaces of the product spaces $\Sf^n\times \R$ and $\Hy^n\times \R$ that have flat normal bundle when regarded as submanifolds with codimension two of the underlying flat spaces $\R^{n+2}\supset…

Consider the Euclidean space $\mathbb{R}^3$ endowed with a canonical semi-symmetric non-metric connection determined by a vector field $\mathsf{C}\in\mathfrak{X}(\mathbb{R}^3)$. We study surfaces when the sectional curvature with respect to…

We consider a generalized angle in complex normed vector spaces. Its definition corresponds to the definition of the well known Euclidean angle in real inner product spaces. Not surprisingly it yields complex values as `angles'. This…

In this study, we give the relationships between the conical curvatures of ruled surfaces drawn by the unit vectors of the ruling, central normal and central tangent of a regular ruled surface in the Euclidean -space. We obtain the…

We will obtain the warped product decompositions of spaces of constant curvature (with arbitrary signature) in their natural models as subsets of pseudo-Euclidean space. This generalizes the corresponding result by S. Nolker to arbitrary…

We consider general relativity with cosmological constant minimally coupled to the electromagnetic field and assume that the four-dimensional space-time manifold is a warped product of two surfaces with Lorentzian and Euclidean signature…

We extend the estimate obtained in [1] for the mean curvature of a cylindrically bounded proper submanifold in a product manifold with an Euclidean space as one factor to a general product ambient space endowed with a warped product…

We study inverse mean curvature flows of starshaped, mean convex hypersurfaces in warped product manifolds with a positive warping factor $\varphi(r)$. If $\varphi'(r)>0$ and $\varphi''(r)\geq 0$, we show that these flows exist for all…

In this article we classify the totally umbilical surfaces which are immersed into a wide class of Riemannian manifolds having a structure of warped product, more precisely, we show that a totally umbilical surface immersed into the warped…

We consider a unit speed curve $\alpha$ in Euclidean four-dimensional space $E^4$ and denote the Frenet frame by $\{T,N,B_1,B_2\}$. We say that $\alpha$ is a slant helix if its principal normal vector $N$ makes a constant angle with a fixed…

We study surfaces in Euclidean space constructed by the sum of two curves or that are graphs of the product of two functions. We consider the problem to determine all these surfaces with constant Gauss curvature. We extend the results to…