arXiv research

The paper classifies conformal solitons in pseudo-Euclidean spaces.

The paper studies special surfaces in pseudo-Euclidean space.

Examines medial axis in pseudo-Euclidean spaces.

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

The paper classifies ruled minimal surfaces in pseudo-Euclidean space.

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…

The paper proves rigidity for self-shrinking spacelike graphs in pseudo-Euclidean space.

New parametrizations for minimal timelike surfaces discovered.

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

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

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

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.

Minimal Lorentz surfaces in pseudo-Euclidean 4-space are characterized by specific curvature conditions.

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…

Reduces gradient Ricci solitons to ODEs for easier analysis.

Study static Einstein-Maxwell space invariant by translation.

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

The paper studies cohomogeneity one actions on pseudo-Euclidean space and identifies unique orbit structures.

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.

The paper proves rigidity of spacelike translating solitons in pseudo-Euclidean space.

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…

The paper provides a Weierstrass representation for maximal space-like surfaces in 4D pseudo-Euclidean space.

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…

Study proves long-term flow for special geometric shapes.

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 study characterizes flat pseudo-Euclidean Lie algebras and their properties.

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.

The paper identifies invariant solutions for gradient Ricci almost solitons.

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…