arXiv research

The paper classifies conformal solitons in pseudo-Euclidean spaces.

Examines medial axis in pseudo-Euclidean spaces.

The paper studies special surfaces in pseudo-Euclidean space.

Study on hypersurfaces in pseudo-Euclidean space with constant curvature or rotational properties.

In this paper we deal with curves with degeneration degree two in pseudo-Euclidean spaces of index two. We characterize Bertrand curves. We show a correspondence between the evolute of a null curve and the involute of a certain spacelike curve in the $6-$dimensional pseudo-Euclidean space of index two. Also we characte…

New parametrizations for minimal timelike surfaces discovered.

Develops weak formulation for spacelike flows in pseudo-Euclidean space.

Conditions, related to the so-called bending problem are considered for hypersurfaces of a pseudo-Euclidean space. Corresponding theorems are proved.

Study light ray transform in pseudo-Euclidean space, derive inversion formula, and prove stability.

New, algebraic surfaces found in curved spaces.

We establish the longtime existence and convergence results of the mean curvature flow of entire Lagrangian graphs in Pseudo-Euclidean space which is related to Logarithmic gradient flow.

The paper applies Clairaut's theorem to rotational surfaces in pseudo-Euclidean 4-space.

We give the classification of constant mean curvature rotational surfaces of elliptic, hyperbolic, and parabolic type in the four-dimensional pseudo-Euclidean space with neutral metric.

The paper classifies invariant gradient $k$-Yamabe solitons in pseudo-Euclidean spaces.

We introduce a class of potential submanifolds in pseudo-Euclidean spaces (each N-dimensional potential submanifold is a special flat torsionless submanifold in a 2N-dimensional pseudo-Euclidean space) and prove that each N-dimensional Frobenius manifold can be locally represented as an N-dimensional potential submanif…

Many classical facts in Riemannian geometry have their pseudo-Riemannian analogs. For instance, the spaces of space-like and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure.…

We introduce a class of k-potential submanifolds in pseudo-Euclidean spaces and prove that for an arbitrary positive integer k and an arbitrary nonnegative integer p, each N-dimensional Frobenius manifold can always be locally realized as an N-dimensional k-potential submanifold in ((k + 1) N + p)-dimensional pseudo-Eu…

Study static Einstein-Maxwell space invariant by translation.

This paper gives, in generic situations, a complete classification of ruled minimal surfaces in pseudo-Euclidean space with arbitrary index. In addition, we discuss the condition for ruled minimal surfaces to exist, and give a counter-example on the problem of Bernstein type.

The paper studies mean curvature flow of Lagrangian graphs in pseudo-Euclidean space.

In this paper, we study general rotational surfaces in the 4- dimensional pseudo-Euclidean space E4-2 and obtain a characterization of flat general rotation surfaces with pointwise 1-type Gauss map in E4-2 and give an example of such surfaces.

In this paper, we firstly establish a new volume growth estimate for spacelike entire graphs in the pseudo-Euclidean space $\mathbb{R}^{m+n}_n$. Then by using this volume growth estimate and the Co-Area formula, we prove various rigidity results for spacelike entire self-shrinking graphs.

In this paper, we study rotational surfaces of elliptic, hyperbolic and parabolic type with pointwise 1-type Gauss map which have spacelike profile curve in four dimensional pseudo Euclidean space E4-2 and obtain some characterizations for these rotational surfaces to have pointwise 1-type Gauss map.

A Lorentz surface in the four-dimensional pseudo-Euclidean space with neutral metric is called quasi-minimal if its mean curvature vector is lightlike at each point. In the present paper we obtain the complete classification of quasi-minimal Lorentz surfaces with pointwise 1-type Gauss map.

The study examines hypersurfaces in pseudo-Euclidean space with specific curvature properties.

We prove that the associativity equations of two-dimensional topological quantum field theories are very natural reductions of the fundamental nonlinear equations of the theory of submanifolds in pseudo-Euclidean spaces and give a natural class of potential flat torsionless submanifolds. We show that all potential flat…

Study on hyperspheres in 4-spaces as special Riemannian manifolds.

In the four-dimensional pseudo-Euclidean space with neutral metric there are three types of rotational surfaces with two-dimensional axis - rotational surfaces of elliptic, hyperbolic or parabolic type. A surface whose mean curvature vector field is lightlike is said to be quasi-minimal. In this paper we classify all q…

In this article we construct L--A representations of geodesic flows on quadrics and of billiard problems within ellipsoids in the pseudo--Euclidean spaces. A geometric interpretation of the integrability analogous to the classical Chasles theorem for symmetric ellipsoids is given. We also consider a generalization of t…

In the present paper we consider a special class of Lorentz surfaces in the four-dimensional pseudo-Euclidean space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with timelike or spacelike axis and call them meridian surfaces. We give the complete classification of minimal…

I classify spacelike self-similar shrinking solutions of the mean curvature flow in pseudo-euclidean space in arbitrary codimension, if the mean curvature vector is not a null vector and the principal normal vector is parallel in the normal bundle. Moreover, I exclude the existence of such self-shrinkers in several cas…

We consider gradient Ricci solitons conformal to a $n$-dimensional pseudo-Euclidean space and we completely describe the most general ansatz that reduces the resulting system of partial differential equations to a system of ordinary differential equations. As a consequence, the gradient Ricci solitons that arise from t…

We define general rotational surfaces of elliptic and hyperbolic type in the pseudo-Euclidean 4-space with neutral metric which are analogous to the general rotational surfaces of C. Moore in the Euclidean 4-space. We study Lorentz general rotational surfaces with plane meridian curves and give the complete classificat…

The unique third-order invariant variational equation in three-dimensional (pseudo)Euclidean space is derived.

In this work, we study some classes of rotational surfaces in the pseudo-Euclidean space $\mathbb{E}^4_t$ with profile curves lying in 2-dimensional planes. First, we determine all such surfaces in the Minkowski 4-space $\mathbb{E}^4_1$ with pointwise 1-type Gauss map of the first kind and second kind. Then, we obtain …

In this work, we study a class of rotational surfaces in the pseudo-Euclidean space $\mathbb{E}_2^4$ whose profile curves lie in two-dimensional planes. We solve the differential equation that characterizes the rotational surfaces with zero mean curvature to determine the profile curves of such rotational surfaces. The…

Study on $ρ$-Einstein solitons with zero scalar curvature, proving stability and flatness.

A flat pseudo-Euclidean Lie algebra is a real Lie algebra with a non degenerate symmetric bilinear form and a left symmetric product whose the commutator is the Lie bracket and such that the left multiplications are skew-symmetric. We show that the center of a flat pseudo-Euclidean nilpotent Lie algebra of signature $(…

The paper studies mean curvature flow of spacelike-convex submanifolds in pseudo-Euclidean space.

A complete system of differential invariants for equivalence of curves in the $n$-dimensional pseudo-euclidean space with respect to the action of each of the groups $K^n \lhd O(n,p,K)$, $K^n \lhd SO(n,p,K)$, $O(n,p,K)$, and $SO(n,p,K)$, where $K = \mathbf{R}$, or $K = \mathbf{C}$, and $O(n,p,K)$ $\bigl($respectively, …

Study on rotating surfaces in 4D space with matrices.

We study warped products semi-Riemannian Einstein manifolds. We consider the case in that the base is conformal to an n-dimensional pseudo Euclidean space and invariant under the action of an translation group. We provide all such solutions in the case Ricci flat when the base is conformal to an n-dimensional pseudo-Eu…

Unified view of surfaces in R^n using Gauss map, caustics, and quadratic forms.

We construct a special class of Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with timelike or spacelike axis and call them meridian surfaces. We give the complete classification of the meridian surfaces with parallel mean c…

We study minimal Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric whose first normal space is two-dimensional and whose Gauss curvature $K$ and normal curvature $\varkappa$ satisfy the inequality $K^2-\varkappa^2 >0$. Such surfaces we call minimal Lorentz surfaces of general type. On any surface of …

We give a loop group formulation for the problem of isometric immersions with flat normal bundle of a simply connected pseudo-Riemannian manifold $M_{c,r}^m$, of dimension $m$, constant sectional curvature $c \neq 0$, and signature $r$, into the pseudo-Euclidean space $\real_s^{m+k}$, of signature $s\geq r$. In fact th…

We prove some Bernstein theorems for entire space-like submanifolds in pseudo-Euclidean spaces and, as a corollary, we obtain a new proof of the Calabi-Pogorelov theorem on global solutions of Monge-Ampere equations.

We find a one-to-one correspondence between full extrinsic symmetric spaces in (possibly degenerate) inner product spaces and certain algebraic objects called (weak) extrinsic symmetric triples. In particular, this yields a description of arbitrary extrinsic symmetric spaces in pseudo-Euclidean spaces by corresponding …