arXiv research

The paper investigates compatible linear connections on Randers spaces and finds a unique extremal connection.

Extending Lévi-Civita's concept to non-quadratic spaces, this study finds extremal compatible linear connections.

The paper finds the extremal compatible linear connection on generalized Berwald manifolds.

The paper characterizes compatible linear connections on 3D Finsler manifolds.

New proof for unique semi-symmetric compatible linear connection on Finsler manifolds.

The paper explores conditions for Randers metrics to have compatible linear connections.

Constructs optimal symplectic connections for Kaehler metrics on holomorphic submersions.

New approach connects Finsler geometry's metric and connections.

Integrable hierarchies linked to F-manifolds with compatible connection.

We discuss and investigate the problem of existence of metric-compatible linear connections for a given space-time metric which is, generally, assumed to be semi-pseudo-Riemannian. We prove that under sufficiently general conditions such connections exist iff the rank and signature of the metric are constant. On this b…

Study of multiplicative connections in Lie groupoids.

I do not agree with the authors of papers arXiv:0806.2184 and arXiv:0901.1023v1 (published in Phys. Lett., respectively, B668 (2008) 453 and B676 (2009) 173). They consider that \textit{"In Finsler manifold, there exists a unique linear connection - the Chern connection ... It is torsion freeness and metric compatibili…

Given a path of almost-Kähler metrics compatible with a fixed symplectic form on a compact 4-manifold such that at time zero the almost-Kähler metric is an extremal Kähler one, we prove, for a short time and under a certain hypothesis, the existence of a smooth family of extremal almost-Kähler metrics compatible with t…

We show that a properly convex projective structure $\mathfrak{p}$ on a closed oriented surface of negative Euler characteristic arises from a Weyl connection if and only if $\mathfrak{p}$ is hyperbolic. We phrase the problem as a non-linear PDE for a Beltrami differential by using that $\mathfrak{p}$ admits a compatib…

The aim of this paper is to study from the point of view of linear connections the data $(M,\mathcal{D},g,W),$ with $M$ a smooth $(n+p)$ dimensional real manifold, $(\mathcal{D},g)$ a \textit{$n$}\textit{\emph{dimensional semi-Riemannian distribution}}\emph{}on $M,$ $\mathcal{G}$ the conformal structure generated by $g…

We formulate an approach to the geometry of Riemann-Cartan spaces provided with nonholonomic distributions defined by generic off-diagonal and nonsymmetric metrics inducing effective nonlinear and affine connections. Such geometries can be modelled by moving nonholonomic frames on (pseudo) Riemannian manifolds and desc…

Simplified proof of foliation closure theorem for linear foliations.

In the paper we present results about generalized Berwald surfaces involving the intrinsic characterization, some topological obstructions for the base manifold and examples.

We call a manifold with torsion and nonmetricity the metric-affine manifold. The nonmetricity leads to a difference between the auto parallel line and the extreme line, and to a change in the expression of the Frenet transport and moving basis. The torsion leads to a change in the Killing equation. We also need to add …

In this contribution we review results on the kinematics of a quantum system localized on a connected configuration manifold and compatible dynamics for the quantum system including external fields and leading to non-linear Schrödinger equations for pure states.

The space of symplectic connections on a symplectic manifold is a symplectic affine space. M. Cahen and S. Gutt showed that the action of the group of Hamiltonian diffeomorphisms on this space is Hamiltonian and calculated the moment map. This is analogous to, but distinct from, the action of Hamiltonian diffeomorphism…

It is well-known that a torsion-free linear connection on a light-like manifold $(M,g)$ compatible with the degenerate metric $g$ exists if and only if $Rad(TM)$ is a Killing distribution. In case of existence, there is an infinitude of connections with none distinguished. We propose a method to single out connections …

Linear F-manifolds are studied with connections and dual spaces.

The paper explores existence and non-existence of constant scalar curvature and extremal Sasaki metrics.

Generalized Lagrange-Weyl structures and compatible connections are introduced as a natural generalization of similar notions from Riemannian geometry. Exactly as in Riemannian case, the compatible connection is unique if certain symmetry conditions with respect to vertical and horizontal Christoffel symbols are impose…

New approach simplifies proof of wave equations on black holes.

New method trains deep vanilla networks as fast as ResNets without shortcut connections.

The paper explores how the Gauss curvature of Riemannian surfaces can be represented as the divergence of a vector field.

The aim of this paper is to adapt the general multitime maximum principle to a Riemannian setting. More precisely, we intend to study geometric optimal control problems constrained by the metric compatibility evolution PDE system; the evolution ("multitime") variables are the local coordinates on a Riemannian manifold,…

We consider the Chern connection of a (conic) pseudo-Finsler manifold $(M,L)$ as a linear connection $\nabla^V$ on any open subset $Ω\subset M$ associated to any vector field $V$ on $Ω$ which is non-zero everywhere. This connection is torsion-free and almost metric compatible with respect to the fundamental tensor $g$.…

In previous work, we have constructed diagrammatic idempotents in an affine extension of the Temperley-Lieb category, which describe extremal weight projectors for sl(2), and which categorify Chebyshev polynomials of the first kind. In this paper, we generalize the construction of extremal weight projectors to the case…

A linear connection on a Lie algebroid is called a Cartan connection if it is suitably compatible with the Lie algebroid structure. Here we show that a smooth connected manifold $M$ is locally homogeneous - i.e., admits an atlas of charts modeled on some homogeneous space $G/H$ - if and only if there exists a transitiv…

Reply to Tetlock et al. on tail risk and probability gap.

Bi-Hamiltonian structures are of great importance in the theory of integrable Hamiltonian systems. The notion of compatibility of symplectic structures is a key aspect of bi-Hamiltonian systems. Because of this, a few different notions of compatibility have been introduced. In this paper we show that, under some additi…

Classifies connections on Galilei manifolds, generalizing known results.

The article classifies liftings of connections on differential manifolds for geodesic modeling.

We consider generic static spacetimes with Killing horizons and study properties of curvature tensors in the horizon limit. It is determined that the Weyl, Ricci, Riemann and Einstein tensors are algebraically special and mutually aligned on the horizon. It is also pointed out that results obtained in the tetrad adjust…

Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors. By the fundamental result of the theory \cite{V5} such a linear connection must be metrical with respect to the averaged Riemannian metric given by integr…

We semiclassicalise the theory of quantum group principal bundles to the level of Poisson geometry. The total space $X$ is a Poisson manifold with Poisson-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with bicovariant Poisson-compatible contravariant connection, and the …

We establish an equivalence between conformally Einstein--Maxwell Kahler 4-manifolds (recently studied in many works) and extremal Kahler 4-manifolds (in the sense of Calabi) with nowhere vanishing scalar curvature. The corresponding pairs of Kahler metrics arise as transversal Kahler structures of Sasaki metrics compa…

An almost Kähler structure is {\it extremal} if the Hermitian scalar curvature is a Killing potential [29]. When the almost complex structure is integrable it coincides with extremal Kähler metric in the sense of Calabi [8]. We observe that the existence of an extremal {\it toric} almost Kähler structure of involutive …

We develop the formalism for noncommutative differential geometry and Riemmannian geometry to take full account of the *-algebra structure on the (possibly noncommutative) coordinate ring and the bimodule structure on the differential forms. We show that *-compatible bimodule connections lead to braid operators $σ$ in …

The paper establishes a correspondence for projective bundles over curves using test configurations and extremal metrics.

The goal of this paper is to encode equivalently the fractional Lagrange dynamics as a nonholonomic almost Kahler geometry. We use the fractional Caputo derivative generalized for nontrivial nonlinear connections (N-connections) originally introduced in Finsler geometry, with further developments in Lagrange and Hamilt…

Symmetric connections that are compatible with semi-Riemannian metrics can be characterized using an existence result for an integral leaf of a (possibly non integrable) distribution. In this paper we give necessary and sufficient conditions for a left-invariant connection on a Lie group to be the Levi-Civita connectio…

An introduction into the theory of locally anisotropic spaces (modelled as vector bundles provided with compatible nonlinear and distinguished linear connections and metric structures and containing as particular cases different types of Kaluza--Klein and/or extensions of Lagrange and Finsler spaces) is presented. The …

The construction of gauge theories beyond the realm of Lie groups and algebras leads one to consider Lie groupoids and algebroids equipped with additional geometrical structures which, for gauge invariance of the construction, need to satisfy particular compatibility conditions. This paper analyzes these compatibilitie…

Complete set of local gauge invariants for Kerr spacetime identified.