相关论文: Description of surfaces associated with Grassmanni…

A class of surfaces-graphs in a Riemannian 3-space with a prescribed projection of one field of principal directions onto a surface $\Pi$ is considered. A problem of determination of such surfaces when both principal curvatures are given…

To study spacelike surfaces of codimension two in the Lorentz-Minkowski space $\Bbb R^{n+1}_1,$ we construct a pair of maps whose values are in $HS_r:=H_+^n(\textbf v,1)\cap \{x_{n+1}=r\},$ called $\textbf n_r^{\pm}$-Gauss maps. It is…

A few formulas and theorems for statistical structures are proved. They deal with various curvatures as well as with metric properties of the cubic form or its covariant derivative. Some of them generalize formulas and theorems known in the…

We study singularities of constant positive Gaussian curvature surfaces and determine the way they bifurcate in generic 1-parameter families of such surfaces. We construct the bifurcations explicitly using loop group methods. Constant…

For a regular curve on a spacelike surface in Lorentz-Minkowski $3$-space, we have a moving frame along the curve which is called a Lorentzian Darboux frame. We introduce five special vector fields along the curve associated to the…

Generalizing the notion of domains of dependence in the Minkowski space, we define and study regular domains in the affine space with respect to a proper convex cone. In dimension three, we show that every proper regular domain is uniquely…

In this paper, we consider minimal graphs in the three-dimensional Riemannian manifold $M\times\mathbb{R}$. We mainly estimate the Gaussian curvature of such surfaces. We consider the minimal disks and minimal graphs bounded by two Jordan…

Weingarten surfaces are those whose principal curvatures satisfy a functional relation, whose set of solutions is called the curvature diagram or the W-diagram of the surface. Making use of the notion of geometric linear momentum of a plane…

Author finds the solutions of the Christoffel problem for open and closed surfaces in Riemannian space. The Christoffel problem is reduced to the problem of construction the continuous G-deformations preserving the sum of principal radii of…

In the work \cite{Laredo} the author shows that every hypersurface in Euclidean space is locally associated to the unit sphere by a sphere congruence, whose radius function $R$ is a geometric invariant of hypersurface. In this paper we…

We review the relation between homotopy algebras of conformal field theory and geometric structures arising in sigma models. In particular we formulate conformal invariance conditions, which in the quasi-classical limit are Einstein…

We get new results (and rederive some know ones) on smooth surfaces in $\mathbb{R}^n$ by unifying several view points into a coherent general view. Namely, we show and use new relations of the evolute (caustic) with the curvature ellipse,…

The Gaussian surface area measure and the Gaussian cone measure for $C$-pseudo-cones are introduced and their corresponding Gaussian Minkowski problem and Gaussian log-Minkowski problem are proposed, respectively. The existence and…

Let $\Sigma$ denote a closed surface with constant mean curvature in $\mathbb{G}^3$, a 3-dimensional Lie group equipped with a bi-invariant metric. For such surfaces, there is a harmonic Gauss map which maps values to the unit sphere within…

We discuss the validity of Minkowski integral identities for hypersurfaces inside a cone, intersecting the boundary of the cone orthogonally. In doing so we correct a formula provided in [3]. Then we study rigidity results for constant mean…

Embedding diagrams prove to be quite useful when learning general relativity as they offer a way of visualizing spacetime curvature through warped two dimensional (2D) surfaces. In this manuscript we present a different 2D construct that…

In this paper, we prove the existence of smooth, entire, strictly convex, spacelike, constant $\sigma_k$ curvature hypersurfaces with prescribed lightlike directions in Minkowski space. This is equivalent to prove the existence of smooth,…

The Gauss map $g$ of a surface $\Sigma$ in $\mathbb{R}^4$ takes its values in the Grassmannian of oriented 2-planes of $\mathbb{R}^4$: $G^+(2,4)$. We give geometric criteria of stability for minimal surfaces in $\mathbb{R}^4$ in terms of…

We consider a special family of 2-dimensional timelike surfaces in the Minkowski 4-space $\mathbb{R}^4_1$ which lie on rotational hypersurfaces with timelike axis and call them meridian surfaces of elliptic type. We study the following…

We address the problem of second order conformal deformation of spacelike surfaces in compactified Minkowski 4-space. We explain the construction of the exterior differential system of conformal deformations and discuss its general and…