Related papers: Surfaces with central cross-sections

Let us have in S^2, R^2 or H^2 a pair of convex bodies, for S^2 different from S^2, such that the intersections of any congruent copies of them are centrally symmetric. Then our bodies are congruent circles. If the intersections of any…

Smooth real cubic surfaces are birationally trivial (over $\R$) if and only if their real locus is connected or, equivalently, if and only if they have two skew real lines or two skew complex conjugate lines. In such a case a…

We present some families of cubic hypersurfaces in $\mathbb P^5 (\mathbb C)$ containing a plane whose associated quadric bundle does not have a rational section.

Surfaces and curves play an important role in geometric design. In recent years, problem of finding a surface passing through a given curve has attracted much interest. In the present paper, we propose a new method to construct a surface…

We find all analytic surfaces in space $\mathbb{R}^3$ such that through each point of the surface one can draw two transversal circular arcs fully contained in the surface. The problem of finding such surfaces traces back to the works of…

A submanifold is said to be tangentially biharmonic if the bitension field of the isometric immersion that defines the submanifold has vanishing tangential component. The purpose of this paper is to prove that a surface in Euclidean…

Surfaces with concentric $K$-contours and parallel $K$-contours in Euclidean $3$-space are defined. Crucial examples are presented and characterization of them are given.

We investigate differential geometric properties of a parabolic point of a surface in the Euclidean three space. We introduce the contact cylindrical surface which is a cylindrical surface having a degenerate contact type with the original…

Define a ``slice'' curve as the intersection of a plane with the surface of a polytope, i.e., a convex polyhedron in three dimensions. We prove that a slice curve develops on a plane without self-intersection. The key tool used is a…

We utilise the two principles of decoupling introduced in [arXiv:2407.16108] to prove decoupling for two types of surfaces exhibiting radial symmetry. The first type are surfaces of revolution in $\mathbb R^n$ generated by smooth surfaces…

In this paper we partially address two issues: - The first is a rigidity property for pairs (S,C) consisting of a general projective K3 surface S, and a curve C obtained as the normalization of a nodal, hyperplane section of S. We prove…

Canonical principal parameters are introduced for surfaces in $\mathbb R^3$ without umbilical points. It is proved that in these parameters the surface is determined (up to position in space) by a pair of invariants satisfying a partial…

Given a non circular spacial closed curve whose total torsion is an integer multiple of $2\pi$, we construct a germ of a smooth surface that contains it as a hyperbolic principal cycle.

This paper examines the relationship between the knotting of an embedded surface in $\R^3$ and the knotting of its fold curves, formed by the singular set of projection to a plane. The first result shows that every surface, no matter how…

We study the {\it arc and curve} complex $AC(S)$ of an oriented connected surface $S$ of finite type with punctures. We show that if the surface is not a sphere with one, two or three punctures nor a torus with one puncture, then the…

Let $M$ be a convex body and let $K$ be a closed convex surface $K$ both contained in the Euclidean space $\mathbb{E}^3$. What can we say about $M$ if $K$ encloses $M$ and if from all the points in $K$ the body $M$ looks the same? In this…

Let C be a curve (possibly non reduced or reducible) lying on a smooth algebraic surface. We show that the canonical ring R(C, \omega_C) is generated in degree 1 if C is numerically 4-connected, not hyperelliptic and even (i.e. with K_C of…

For a finite planar graph, it associates with some metric spaces, called (regular) spherical polyhedral surfaces, by replacing faces with regular spherical polygons in the unit sphere and gluing them edge-to-edge. We consider the class of…

We obtain an infinite family of complete non embedded rotational surfaces in $\mathbb R^3$ whose second fundamental forms have length equal to one at any point. Also we prove that a complete rotational surface with second fundamental form…

We show that a capillary surface in a solid cone, that is, a surface that has constant mean curvature and the boundary of surface meets the boundary of the cone with a constant angle, is radially graphical if the mean curvature is…