arXiv research

The paper discusses Gauss maps for Möbius surfaces in spheres and their applications to Willmore surfaces.

The goal of this paper, using lifting theory it is to produce almost paracomplex struc- tures on the tangent bundle of almost Lorentzian r-paracontact manifold endowed with almost Lorentzian r-paracontact structure. Finally, we discuss the effect over dynamics systems of the produced geometrical structures.

In this study, taking into considering lifting theory, we shall obtain both almost complex and paracomplex structures on the tangent bun- dle, based on almost Lorentzian r-contact and r-paracontact manifold.

Unified approach to constructing integrable systems using Stäckel lifts.

Based on the Hamiltonian dimensional reduction of $3+1$ axially symmetric, Ricci-flat Lorentzian spacetimes to a $2+1$ Einstein-wave map system with the (negatively curved) hyperbolic 2-plane target, we construct a positive-definite, (spacetime) gauge-invariant energy functional for linear axially symmetric perturbatio…

New framework for generalized Killing spinors on manifolds.

Study on Hausdorff dimension and curvature bounds in sub-Lorentzian Heisenberg group.

Novel geodesic results on affine and Lorentzian manifolds.

The paper constructs foliations of minimal surfaces in negatively curved 3-manifolds.

Geometric structures on surfaces relate to 2-plane distributions in 5D.

Develops a lifting theory for exponential maps in semi-Riemannian geometry.

New methods for constructing null fluid metrics and solving optical lift conjectures.

A diffeomorphism of pseudo-Riemannian manifolds is called sectional curvature preserving if it preserves the sectional curvature of all the nondegenerate 2-planes. We consider a similar condition for degenerate 2-planes and we prove that the diffeomorphism is conformal (when the condition is fulfilled for weakly degene…

The paper analyzes equations for surfaces in 4D space forms.

It is shown that coassociative cones in R^7 that are r-oriented and ruled by 2-planes are equivalent to CR-holomorphic curves in the oriented Grassmanian of 2-planes in R^7. The geometry of these CR-holomorphic curves is studied and related to holomorphic curves in S^6. This leads to an equivalence between associative …

The axiom of θ-holomorphic 2-planes is introduced. It is proved, that if an almost Hermitian manifold satisfies this axiom for a fixed θ, 0< θ< π/2, then it is a real space form.

A maximal surface $\sb$ with isolated singularities in a complete flat Lorentzian 3-manifold $\N$ is said to be entire if it lifts to a (periodic) entire multigraph $\tilde{\sb}$ in $ł^3.$ In addition, $\sb$ is called of finite type if it has finite topology, finitely many singular points and $\tilde{\sb}$ is finitely …

Cayley cones in the octonions $\mathbb{O}$ that are ruled by oriented 2-planes are equivalent to pseudoholomorphic curves in the Grassmannian of oriented 2-planes G(2,8). The well known twistor fibration $G(2,8) -> S^6$ is used to prove the existence of immersed higher-genus pseudoholomorphic curves in $\gro$. Equivale…

We study the geometry of Engel structures, which are 2-plane fields on 4-manifolds satisfying a generic condition, that are compatible with other geometric structures. A \em{Lagrangian} Engel structure is an Engel 2-plane field on a symplectic 4-manifold for which the 2-planes are Lagrangian with respect to the symplec…

The paper proves a Minkowski-like theorem for tetrahedra in dS3 and AdS3.

Study contact structures on projective spaces, proving infinite non-isotopic structures.

By referring to theorems of Donaldson and Hitchin, we exhibit a rigorous AdS/CFT-type correspondence between classical 2+1 dimensional vacuum general relativity theory on S x R and SO(3) Hitchin theory (regarded as a classical conformal field theory) on the spacelike past boundary S, a compact, oriented Riemann surface…

We study the geometry of Engel structures, which are 2-plane fields on 4-manifolds satisfying a generic condition, that are compatible with other geometric structures. A complex Engel structure is an Engel 2-plane field on a complex surface for which the 2-planes are complex lines. We solve the equivalence problems for…

The Grassmannian $V_2(\mathbb{R}^{n+2})$ of oriented 2-planes in $\mathbb R^{n+2}$ where $n\ge3$ carries a homogeneous parabolic contact structure of Grassmannian type. The main result of this article is that on $V_2(\mathbb{R}^{n+2})$ lives an elliptic complex of invariant differential operators of length 3 which star…

For a closed oriented 3-manifold Y, we define an absolute grading on the Heegaard Floer homology groups of Y by homotopy classes of oriented 2-plane fields. We show that this absolute grading refines the relative one and that it is compatible with the maps induced by cobordisms. We also prove that if ξ is a contact str…

Study geometric isomorphisms between spacetime solutions using paracausal metrics.

In some recent papers, the relations existing between the metric properties of Randers spaces and the conformal geometry of stationary Lorentzian manifolds were discovered and investigated. In this note, we focus on the equality between the index of a geodesic in a Randers space and that of its lightlike lift in the as…

The phase space of relativistic particle mechanics is defined as the 1st jet space of motions regarded as timelike 1-dimensional submanifolds of spacetime. A Lorentzian metric and an electromagnetic 2-form define naturally on the odd-dimensional phase space a generalized contact structure. In the paper infinitesimal sy…

A Riemannian manifold is said to be almost positively curved if the sets of points for which all $2$-planes have positive sectional curvature is open and dense. We show that the Grassmannian of oriented $2$-planes in $\mathbb{R}^7$ admits a metric of almost positive curvature, giving the first example of an almost posi…

Proves conjecture about geodesic foliations in Riemannian planes.

In 1998, R. Gompf defined a homotopy invariant $θ_G$ of oriented 2-plane fields in 3-manifolds. This invariant is defined for oriented 2-plane fields $ξ$ in a closed oriented 3-manifold $M$ when the first Chern class $c_1(ξ)$ is a torsion element of $H^2(M;\mathbb{Z})$. In this article, we define an extension of the Go…

Maps from 2-planes to projective spaces using quaternions and octonions.

The `observer space' of a Lorentzian spacetime is the space of future-timelike unit tangent vectors. Using Cartan geometry, we first study the structure a given spacetime induces on its observer space, then use this to define abstract observer space geometries for which no underlying spacetime is assumed. We propose ta…

Inspired by an article of R. Bryant on holomorphic immersions of unit disks into Lorentzian CR manifolds, we discuss the application of Cartan's method to the question of the existence of bi-disk $\mathbb{D}^{2}$ in a smooth $9$-dimensional real analytic real hypersurface $M^{9}\subset\mathbb{C}^{5}$ with Levi signatur…

Nurowski showed that any generic 2-plane field $D$ on a 5-manifold $M$ determines a natural conformal structure $c_D$ on $M$; these conformal structures are exactly those (on oriented $M$) whose normal conformal holonomy is contained in the (split, real) simple Lie group $G_2$. Graham and Willse showed that for real-an…

The Whitney-Graustein theorem states that regular closed curves in the 2-plane are classified, up to regular homotopy, by their rotation number. Here we give a simple proof based on contact geometry.

Given a generic 2-plane field on a 5-dimensional manifold we consider its (3,2)-signature conformal metric [g] as defined in math.DG/0406400. Every conformal class [g] obtained in this way has very special conformal holonomy: it must be contained in the split-real-form of the exceptional group G_2. In this note we show…

Researchers found a way to measure energy in black hole perturbations.

The abstract discusses metrics with positive biorthogonal curvature on 5-manifolds.

In this article we study minimal flat Lorentzian surfaces in Lorentzian complex space forms. First we prove that, for minimal flat Lorentzian surfaces in a Lorentzian complex form, the equation of Ricci is a consequence of the equations of Gauss and Codazzi. Then we classify minimal flat Lorentzian surfaces in the Lore…

The holonomy of the ambient metrics of Nurowski's conformal structures associated to generic real-analytic 2-plane fields on 5-manifolds is investigated. It is shown that the holonomy is always contained in the split real form G_2 of the exceptional Lie group, and is equal to G_2 for an open dense set of 2-plane fields…

We recreate an unpublished proof of William Thurston from the early 1970's that any smooth 2-plane field on a manifold of dimension at least 4 is homotopic to the tangent plane field of a foliation.

In "The Gel'fand-Kalinin-Fuks class and characteristic classes of transversely symplectic foliations", arXiv:0910.3414, (October 2009) by D.Kotschick and S.Morita, the relative Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields without constant vector fields on 2n-plane were characterized b…

Defines metrics for Lorentzian spaces and explores maximal developments.

Study of a series of Lorentzian structures on SL(2,R) with SO(1,1) symmetry.

Study left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.

It is shown how one can apply the classification of the holonomy algebras of Lorentzian manifolds to solve some problems. In particular, a new proof to the classification of Lorentzian manifolds with recurrent curvature tensor is given; the classification of two-symmetric Lorentzian manifolds is explained; conformally …

Conditions, related to Kulkarni's equivalence problem are considered for indefinite Riemannian and Kaehlerian manifolds. Corresponding theorems are obtained for the values of the Ricci tensor on isotropic vectors as well as for the values of the curvature tensor on degenerate holomorphic 2-planes.