arXiv research

The paper establishes a connection between force-free fields and conformally geodesic fields.

A geodesic circle in Finsler geometry is a natural extension of that in a Euclidean space. In this paper, we apply Lie derivatives and the Cartan $Y$-connection to study geodesic circles and (infinitesimal) concircular transformations on a Finsler manifold. We characterize a concircular vector field with some PDEs on t…

Paper proves geodesics and focal points unchanged by conformal changes in pseudo-Finsler manifolds.

We study the integrability of the conformal geodesic flow (also known as the conformal circle flow) on the $SO(3)$--invariant gravitational instantons. On a hyper--Kähler four--manifold the conformal geodesic equations reduce to geodesic equations of a charged particle moving in a constant self--dual magnetic field. In…

The paper studies a new submersion type with specific conditions.

Study foliations on Riemannian manifolds with specific vector fields, focusing on geometric properties.

The paper proves a rigidity theorem for null geodesic travel times in asymptotically AdS spacetimes.

Geodesic vector fields on flat 3-manifolds are related to contact structures.

Some results related to the causality of compact Lorentzian manifolds are proven: (1) any compact Lorentzian manifold which admits a timelike conformal vector field is totally vicious, and (2) a compact Lorentzian manifold covered regularly by a globally hyperbolic spacetime admits a timelike closed geodesic, if some n…

In this paper we examine different aspects of the geometry of closed conformal vector fields on Riemannian manifolds. We begin by getting obstructions to the existence of closed conformal and nonparallel vector fields on complete manifolds with nonpositive Ricci curvature, thus generalizing a theorem of T. K. Pan. Then…

The present note deals with the dynamics of metric connections with vectorial torsion, as already described by E. Cartan in 1925. We show that the geodesics of metric connections with vectorial torsion defined by gradient vector fields coincide with the Levi-Civita geodesics of a conformally equivalent metric. By pullb…

Moitvated in part by [3], in this note we obtain a rigidity result for globally hyperbolic vacuum spacetimes in arbitrary dimension that admit a timelike conformal Killing vector field. Specifically, we show that if M is a Ricci flat, timelike geodesically complete spacetime with compact Cauchy surfaces that admits a t…

We study the Jones and Tod correspondence between selfdual conformal 4-manifolds with a conformal vector field and abelian monopoles on Einstein-Weyl 3-manifolds, and prove that invariant complex structures correspond to shear-free geodesic congruences. Such congruences exist in abundance and so provide a tool for cons…

Quantizes geodesic lengths in Teichmüller spaces using algebraic methods.

Study null geodesics on even-dimensional conformal manifolds, finding Einstein metrics and CR structures.

In this paper we consider the lens rigidity problem with partial data for conformal metrics in the presence of a magnetic field on a compact manifold of dimension $\geq 3$ with boundary. We show that one can uniquely determine the conformal factor and the magnetic field near a strictly convex (with respect to the magne…

We show that Killing tensors on conformally flat $n$-dimensional tori whose conformal factor only depends on one variable, are polynomials in the metric and in the Killing vector fields. In other words, every first integral of the geodesic flow polynomial in the momenta on the sphere bundle of such a torus is linear in…

We show that if a connected compact kählerian surface $M$ with nonpositive gaussian curvature is furnished with a closed conformal vector field $ξ$ whose singular points are isolated, then $M$ is isometric to a flat torus and $ξ$ is parallel. We also consider the case of a connected complete kählerian manifod $M$ of co…

Conformal geodesics can't spiral in Riemannian manifolds.

In this paper, we study the convergence of Yang-Mills-Higgs fields defined on fiber bundles over Riemann surfaces where the fiber is a compact symplectic manifold and the conformal structure of the Riemann surface is allowed to vary. We show that away from the nodes, the YMH fields converges, up to gauge, to a smooth Y…

Akyol M.A. [Conformal anti-invariant submersions from cosymplectic manifolds, Hacettepe Journal of Mathematics and Statistic, 46(2), (2017), 177-192.] defined and studied conformal anti-invariant submersions from cosymplectic manifolds. The aim of the present paper is to define and study the notion of conformal slant s…

We study complex 4-manifolds with holomorphic self-dual conformal structures, and we obtain an interpretation of the Weyl tensor of such a manifold as the projective curvature of a field of cones on the ambitwistor space. In particular, its vanishing is implied by the existence of some compact, simply-connected, null-g…

Study of 3D trans-Sasakian manifolds using Newman--Penrose formalism.

The study characterizes Riemannian manifolds with conformal vector fields and proves isometric properties.

Three explicit families of spacelike Zoll surface admitting a Killing field are provided. It allows to prove the existence of spacelike Zoll surface not smoothly conformal to a cover of de Sitter space as well as the existence of Lorentzian Möbius strips of non constant curvature all of whose spacelike geodesics are cl…

Research proves the semi-classical limit of Liouville conformal field theory, describing deterministic geometry from random fluctuations.

We review properties of so-called special conformal Killing tensors on a Riemannian manifold $(Q,g)$ and the way they give rise to a Poisson-Nijenhuis structure on the tangent bundle $TQ$. We then address the question of generalizing this concept to a Finsler space, where the metric tensor field comes from a regular La…

The connected components of the zero set of any conformal vector field $v$, in a pseudo-Riemannian manifold $(M,g)$ of arbitrary signature, are of two types, which may be called `essential' and `nonessential'. The former consist of points at which $v$ is essential, that is, cannot be turned into a Killing field by a lo…

In 3D, conformal geodesics are variational.

The Fefferman metric connects CR manifolds to conformal geodesics in 3D.

We show that the index of a lightlike geodesic in a conformally standard stationary spacetime is equal to the index of its spatial projection as a geodesic of a Finsler metric associated to the spacetime. Moreover we obtain the Morse relations of lightlike geodesics connecting a point to an integral line of the standar…

We consider a closed orientable Riemannian 3-manifold $(M,g)$ and a vector field $X$ with unit norm whose integral curves are geodesics of $g$. Any such vector field determines naturally a 2-plane bundle contained in the kernel of the contact form of the geodesic flow of $g$. We study when this 2-plane bundle remains i…

Study geodesics in conformally compact manifolds, showing smoothness and asymptotic behavior.

We describe the local conformal geometry of a Lorentzian spin manifold $(M,g)$ admitting a twistor spinor $φ$ with zero. Moreover, we describe the shape of the zero set of $φ$. If $φ$ has isolated zeros then the metric $g$ is locally conformally equivalent to a static monopole. In the other case the zero set consists o…

In this paper we have obtained evolution of some geometric quantities on a compact Riemannian manifold $M^n$ when the metric is a Yamabe soliton. Using these quantities we have obtained bound on the soliton constant. We have proved that the commutator of two soliton vector fields with the same metric in a given conform…

Wojciech Kamiński disproved a spiral claim for conformal geodesics.

New variational principles found for conformal geodesics.

We address the problem of finding conditions under which a compact Lorentzian manifold is geodesically complete, a property, which always holds for compact Riemannian manifolds. It is known that a compact Lorentzian manifold is geodesically complete if it is homogeneous, or has constant curvature, or admits a time-like…

Variationality of conformal geodesics fails in higher dimensions.

Researchers found spiraling conformal geodesics in 3D space.

We show that a conformal connection on a closed oriented surface $Σ$ of negative Euler characteristic preserves precisely one conformal structure and is furthermore uniquely determined by its unparametrised geodesics. As a corollary it follows that the unparametrised geodesics of a Riemannian metric on $Σ$ determine th…

Study peels tensor equations on Schwarzschild spacetime.

This study is motivated by the researches in the field of invariants of geodesic and conformal mappings presented in (T. Y. Thomas, [22]) and (H. Weyl, [25]). The Thomas projective parameter and the Weyl projective tensor are generalized in this article. Generators for vector spaces of invariants of geometric mappings …

Solves Dirichlet problem for harmonic maps to give geodesic insights.

The Schwarzian derivative is generalized to Finsler manifolds and its properties studied.

We consider integral geometry inverse problems for unitary connections and skew-Hermitian Higgs fields on manifolds with negative sectional curvature. The results apply to manifolds in any dimension, with or without boundary, and also in the presence of trapped geodesics. In the boundary case, we show injectivity of th…

With the aid of concrete examples, we consider the question of whether, in the presence of conformal curvature, a conformal geodesic can become trapped in smaller and smaller sets, or phrased informally: are spirals possible? We do not arrive at a definitive answer, but we are able to find situations where this behavio…

We answer to the question whether a system of the 3rd order ODEs describes geodesics of a conformal structure. We construct a functor from a category of conformal geometries to a category of Cartan geometries associated to the 3rd order ODEs systems. Explicit formulas which define the family of all equations on conform…