Related papers: Concircular hypersurfaces and concircular helices …
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.
The existence of closed hypersurfaces of prescribed scalar curvature in globally hyperbolic Lorentzian manifolds is proved provided there are barriers.
We complete the classification of hyperelliptic threefolds, describing in an elementary way the hyperelliptic threefolds with group $D_4$. These are algebraic and form an irreducible 2-dimensional family. Our paper is fully self-contained.
We initiate the study of convex geometry over ordered hyperfields. We define convex sets and halfspaces over ordered hyperfields, presenting structure theorems over hyperfields arising as quotients of fields. We prove hyperfield analogues…
We construct geometrically infinite hyperbolic surfaces supporting horocycles with tailored recurrence properties. In particular, we obtain the first examples of non-trivial minimal horocyclic orbit closures and of infinite locally-finite…
We classify isoparametric hypersurfaces in complex hyperbolic spaces.
In this work we study spacelike hypersurfaces immersed in spatially open standard static spacetimes with complete spacelike slices. Under appropriate lower bounds on the Ricci curvature of the spacetime in directions tangent to the slices,…
In this paper, we study generic conformally flat hypersurfaces in the Euclidean $4$-space $\mathbb{R}^4$ using the framework of M\"{o}bius geometry. First, we classify locally the generic conformally flat hypersurfaces with closed M\"obius…
Using geodesic length functions, we define a natural family of real codimension 1 subvarieties of Teichm\"uller space, namely the subsets where the lengths of two distinct simple closed geodesics are of equal length. We investigate the…
We propose a notion of discrete elastic and area-constrained elastic curves in 2-dimensional space forms. Our definition extends the well-known discrete Euclidean curvature equation to space forms and reflects various geometric properties…
It is pointed out that the current form of extrinsic equation of motion for a particle constrained to remain on a hypersurface is in fact a half-finished version for it is established without regard to the fact that the particle can never…
In this paper, we study the problem of finding a hypersurface family from a given spatial geodesic curve in R4. We obtain the parametric representation for a hypersurface family whose members have the same curve as a given geodesic curve.…
The hypersurface is one of the most important objects in a space. Many authors studied diffrent geometric aspects of hypersurfaces in a space. In this paper, we define three types of 2-ruled hypersurfaces in a Walker 4-manfold E 41 . We…
In an inviscid and incompressible fluid in dimension 3, we prove the existence of several helical filaments, or vortex helices, collapsing into each others.
We give a short proof of the contractibility of the space of geodesic triangulations with fixed combinatorial type of a convex polygon in the Euclidean plane. Moreover, for any $n>0$, we show that there exists a space of geodesic…
We study area-stationary, or maximal, surfaces in the space ${\mathbb L}({\mathbb H}^3)$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral K\"ahler structure. We prove that every holomorphic curve in ${\mathbb…
We define twisted Alexander polynomials of a complex hypersurface with arbitrary singularities. These generalize the classical Alexander polynomials of high dimensional hypersurfaces and the twisted Alexander polynomial of plane curves. We…
We study the closure of horocycles on rank 1 nonpositively curved surfaces with finitely generated fundamental group. Each horocycle is closed or dense on a certain subset of the unit tangent bundle. In fact, we classify each half-horocycle…
In this paper we describe trigonometry on the de Sitter surface. For that a characterization of geodesics is given, leading to various types of triangles. We define lengths and angles of these. Then, transferring the concept of polar…
We investigate surfaces with constant harmonic-mean curvature one (HMC-1 surfaces) in hyperbolic three-space. We allow them to have certain kinds of singularities, and discuss some global properties. As well as flat surfaces and surfaces…