arXiv research

We completely classify the algebraic Ricci solitons of four-dimensional pseudo-Riemannian generalized symmetric spaces.

We give a complete description of semi-symmetric algebraic curvature tensors on a four-dimensional Lorentzian vector space and we use this description to determine all four-dimensional homogeneous semi-symmetric Lorentzian manifolds.

We classify non-reductive four-dimensional homogeneous conformally Einstein manifolds.

Paper proves biharmonic hypersurfaces in nonzero space form have constant mean curvature.

The problem of conformal transformation and conformal flatness of Finsler spaces has been studied by so many researchers $\left[ 6,16,17,20,21\right] .$ Recently, Prasad et. al $\left[ 19\right]$ have studied three dimensional conformally flat Landsberg and Berwald spaces and have given some important results. The pur…

The spaces of harmonic maps of the projective plane to the four-dimensional sphere are investigated in this paper by means of twistor lifts. It is shown that such spaces are empty in case of even harmonic degree. In case of harmonic degree less than 6 it was shown that such spaces are path-connected and an explicit par…

In this paper, based on the theory of surfaces in the four-dimensional Euclidean space which generalizes the theory of surfaces in three-dimensional Euclidean space, beside other results, we will give a characterization of points on particular kind of these surfaces.

A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. We associate a geometrically determined moving frame field to such a surface and using the derivative formulas for this frame field we obtain seven invariant functions. Our…

The paper studies Gauss maps of space-like stationary surfaces in Lorentz-Minkowski space, focusing on ramification and unicity.

We classify the periodic Hamiltonian flows on compact four dimensional symplectic manifolds up to isomorphism of Hamiltonian S^1 spaces. Additionally, we show that all these spaces are Kaehler, that every such space is obtained from a simple model by a sequence of symplectic blowups, and that if the fixed points are is…

Quasiclassical generalized Weierstrass representation (GWR) for highly corrugated surfaces with slow modulation in the four-dimensional Euclidean space is proposed. Integrable deformations of such surfaces are described by the dispersionless Davey-Stewartson hierarchy. Quasiclassical GWRs for other four-dimensional spa…

We consider four dimensional conformally flat homogeneous pseudo Riemannian manifolds. According to forms (Seger types) of the Ricci operator, we provide a full classification of four dimensional pseudo Riemannian conformally flat homogeneous Ricci solitons.

We obtain an equivariant classification for orientable, closed, four-dimensional Alexandrov spaces admitting an isometric torus action. This generalizes the equivariant classification of Orlik and Raymond of closed four-dimensional manifolds with torus actions. Moreover, we show that such Alexandrov spaces are equivari…

We classify four-dimensional compact solvmanifolds up to diffeomorphism, while determining which of them have complex analytic structures. In particular, we shall see that a four-dimensional compact solvmanifold S can be written, up to double covering, as G/L where G is a simply connected solvable Lie group and L is a …

We consider three- and four-dimensional pseudo-Riemannian generalized symmetric spaces, whose invariant metrics were explicitly described in [15]. While four-dimensional pseudo-Riemannian generalized symmetric spaces of types A, C and D are algebraic Ricci solitons, the ones of type B are not so. The Ricci soliton equa…

Minimal surfaces found in 4D space.

We provide an alternative method for obtaining of compatible Poisson structures on Lie groups by means of the adjoint representations of Lie algebras. In this way, we calculate some compatible Poisson structures on four dimensional and nilpotent six dimensional symplectic real Lie groups. Then using Magri-Morosi's theo…

We prove that an isometric immersion of a timelike surface in four-dimensional Minkowski space is equivalent to a normalized spinor field which is a solution of a Dirac equation on the surface. Using the quaternions and the complex numbers, we obtain a spinor representation formula that relates the spinor field and the…

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…

A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. We introduce meridian surfaces of parabolic type as one-parameter systems of meridians of a rotational hypersurface with lightlike axis in Minkowski 4-space and find their …

Survey on four-dimensional Thurston geometries with Riemannian metrics.

Classifies 4D spaces with compact Clifford-Klein forms.

New parametrizations for minimal timelike surfaces discovered.

We consider spacelike surfaces in the four-dimensional Minkowski space and introduce geometrically an invariant linear map of Weingarten-type in the tangent plane at any point of the surface under consideration. This allows us to introduce principal lines and an invariant moving frame field. Writing derivative formulas…

General rotational surfaces as a source of examples of surfaces in the four-dimensional Euclidean space have been introduced by C. Moore. In this paper we consider the analogue of these surfaces in the Minkowski 4-space. On the base of our invariant theory of spacelike surfaces we study general rotational surfaces with…

We study the class of spacelike surfaces in the four-dimensional Minkowski space whose mean curvature vector at any point is a non-zero spacelike vector or timelike vector. These surfaces are determined up to a motion by eight invariant functions satisfying some natural conditions. The subclass of Chen surfaces is char…

We classify the non-degenerate homogeneous hypersurfaces in real and complex affine four-space whose symmetry group is at least four-dimensional.

The paper examines 4D hypersurfaces with constant mean curvature in pseudo-Riemannian space forms.

We conclude the construction of the algebraic complex, consisting of spaces of differentials of Euclidean metric values, for four-dimensional piecewise-linear manifolds. Assuming that the complex is acyclic, we investigate how its torsion changes under rebuildings of the manifold triangulation. First, we write out form…

Study finds all 4D neutral manifolds.

New types of Ricci solitons found in 4D Lorentzian geometry.

At any point of a surface in the four-dimensional Euclidean space we consider the geometric configuration consisting of two figures: the tangent indicatrix, which is a conic in the tangent plane, and the normal curvature ellipse. We show that the basic geometric classes of surfaces in the four-dimensional Euclidean spa…

In this paper we prove $L^{\infty}$ type gap theorems in Yang-Mills theory for complete four-dimensional manifolds with a weighted Poincaré inequality. We apply the theorems to a broad class of complete manifolds satisfying weighted Poincaré inequalities. In particular, we obtain a gap theorem on the Euclidean space wi…

We discuss the four-dimensional massless spectrum of supersymmetric Minkowski compactifications of ten-dimensional heterotic supergravity, including the anomaly cancelation condition. This can be calculated from restrictions arising from F-term conditions in a four-dimensional effective theory. The results agree with c…

We consider four dimensional lie groups equipped with left invariant Lorentzian Einstein metrics, and determine the harmonicity properties of vector fields on these spaces. In some cases, all these vector fields are critical points for the energy functional restricted to vector fields. We also classify vector fields de…

The study examines four-dimensional gradient Ricci solitons and their properties.

Given a constant vector field $Z$ in Minkowski space, a timelike surface is said to have a canonical null direction with respect to $Z$ if the projection of $Z$ on the tangent space of the surface gives a lightlike vector field. In this paper we describe these surfaces in the ruled case. For example when the Minkowski …

The paper studies hypersurfaces in 5D space forms with topological and rigidity results.

We write out some sequences of linear maps of vector spaces with fixed bases. Each term of a sequence is a linear space of differentials of metric values ascribed to the elements of a simplicial complex - a triangulation of a manifold. If the sequence turns out to be an acyclic complex then one can construct a manifold…

In four-dimensional gauge theory there exists a well-known correspondence between instantons and holomorphic curves, and a similar correspondence exists between certain octonionic instantons and triholomorphic curves. We prove that this latter correspondence stems from the dynamics of various dimensional reductions of …

It is shown that four-dimensional generalized symmetric spaces can be naturally equipped with some additional structures defined by means of their curvature operators. As an application, those structures are used to characterize generalized symmetric spaces.

We discuss some aspects of the differential geometry of curves in Minkowski space. We establish the Serret-Frenet equations in Minkowski space and use them to give a very simple proof of the fundamental theorem of curves in Minkowski space. We also state and prove two other theorems which represent Minkowskian versions…

We give a physical explanation of the Kontsevich-Soibelman wall-crossing formula for the BPS spectrum in Seiberg-Witten theories. In the process we give an exact description of the BPS instanton corrections to the hyperkahler metric of the moduli space of the theory on R^3 x S^1. The wall-crossing formula reduces to th…

In this paper we prove the path connectedness of the moduli spaces of metrics with positive isotropic curvature on certain compact four-dimensional manifolds.

The study proves non-orientable surfaces can map to a torus.

In this expository paper we present short simple proofs of Conway-Gordon-Sachs' theorem on intrinsic linking in three-dimensional space, as well as van Kampen-Flores' and Ummel's theorems on intrinsic intersections. The latter are related to nonrealizability of certain hypergraphs in four-dimensional space. The proofs …

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.

Study sharp geometric and topological properties of pinched 4D submanifolds.