arXiv research

Study left invariant spray structures on Lie groups, calculating curvature and geodesics.

Geodesic orbit and weakly symmetric properties in spray geometry.

In this paper for a given Banach, possibly infinite dimensional, manifold $M$ we focus on the geometry of its iterated tangent bundle $T^rM$, $r\in {\N}\cup\{\infty\}$. First we endow $T^rM$ with a canonical atlas using that of $M$. Then the concepts of vertical and complete lifts for functions and vector fields on $T^…

The paper establishes conditions for Riemannian connections and semi-simplicity of Lie algebras using spray structures.

This paper classifies geodesics of projectively flat sprays and introduces a method to determine sprays based on geodesics.

Characterizes affine vector fields on Finsler manifolds with rigidity results.

Directly constructs Lie groupoids from Lie algebroids.

The geodesic flow on the tangent bundle is the flow of a certain vector field which is called the spray $S:TM\to TTM$. The flow lines of the vector field $\ka_{TM}øTS:TTM\to TTTM$ project to the Jacobi fields on $TM$. This could be called the Jacobi flow.

Introduces submersion in spray geometry and defines key components.

The paper studies Jacobi fields and conjugate points in projective sprays.

Applying concepts and tools from classical tangent bundle geometry and using the apparatus of the calculus along the tangent bundle projection ('pull-back formalism'), first we enrich the known lists of the characterizations of affine vector fields on a spray manifold and conformal vector fields on a Finsler manifold. …

The paper examines parallel one forms on Riemannian and Finslerian manifolds.

The pseudo-Finsleroid relativistic metric was constructed upon assuming that the involved vector field $b_i$ is time-like. In the present paper it is shown that the metric admits just the alternative counterpart in which the field is space-like. The entailed pseudo-Finsleroid-spatial framework is systematically describ…

This paper examines a generalized Kropina metric and its geometric properties.

From a spray space $S$ on a manifold $M$ we construct a new geometric space $P$ of larger dimension with the following properties: 1. Geodesics in $P$ are in one-to-one correspondence with parallel Jacobi fields of $M$. 2. $P$ is complete if and only if $S$ is complete. 3. If two geodesics in $P$ meet at one point, the…

We prove a monodromy theorem for local vector fields belonging to a sheaf satisfying the unique continuation property. In particular, in the case of admissible regular sheaves of local fields defined on a simply connected manifold, we obtain a global extension result for every local field of the sheaf. This generalizes…

Research explores Lie algebras in Riemannian manifolds.

The projective Finsler metrizability problem deals with the question whether a projective-equivalence class of sprays is the geodesic class of a (locally or globally defined) Finsler function. In this paper we use Hilbert-type forms to state a number of different ways of specifying necessary and sufficient conditions f…

Study of multiplicative connections in Lie groupoids.

The paper studies complex Finsler metrics on complex Lie groups.

The paper studies sprays on Hamel-Funk functions and their properties.

Study shows zero Pontrjagin classes for sprays.

New sprays of constant curvature introduced; conditions for metrizability given.

Defines projective Ricci curvature and proves rigidity for sprays.

Introduces new geodesic fields for Finsler manifolds.

Sprays with vanishing X-curvature are studied in this paper.

We generalize Matsumoto metrics with a special π-form and explore their geometric properties.

Sprays get Hamiltonian description using Dirac structures.

Sprays on Frechet manifolds connect connections and tangent structures.

The paper introduces a new spray deformation and finds new metrics.

The paper classifies spherically symmetric sprays and their curvature properties.

In this paper are constructed a series of geometrical objects on the 1-jet fibre bundle $J^1(T,M)$, which is a basic object in the study of classical and quantum field.

This paper is concerned with the problem of determining whether a projective-equivalence class of sprays is the geodesic class of a Finsler function. We address both the local and the global aspects of this problem. We present our results entirely in terms of a multiplier, that is, a type (0,2) tensor field along the t…

Spray theory aids geometric hydrodynamics methods.

The paper classifies Landsberg metrics on a 2D Lie group and proves a conjecture.

Study reformulates Finsler metrizability problems using geodesic invariance.

In this paper we investigate the following question: under what conditions can a second-order homogeneous ordinary differential equation (spray) be the geodesic equation of a Finsler space. We show that the Euler-Lagrange partial differential system on the energy function can be reduced to a first order system on this …

The main purposes of this article are to extend our previous results on homogeneous sprays to arbitrary (generalized) sprays, to show that locally diffeomorphic exponential maps can be defined for any (generalized) spray, and to give a (possibly nonlinear) covariant derivative for any (possibly nonlinear) connection. I…

Spray-invariant sets maintain geodesics on infinite-dimensional manifolds.

We give a new and complete proof of the following theorem, discovered by Detlef Laugwitz: (forward) complete and connected finite dimensional Finsler manifolds admitting a proper homothety are Minkowski vector spaces. More precisely, we show that under these hypotheses the Finsler manifold is isometric to the tangent M…

The paper characterizes strong Hamel functions using symmetries and proves their preservation properties.

Study left invariant spray geometry on Lie groups using parallel translations.

New method to determine if a spray can be Finslerian.

Let $M$ be a smooth manifold and $\mathcal{S}$ a semi-spray defined on a sub-bundle $\mathcal{C}$ of the tangent bundle $TM$. In this work it is proved that the only non-trivial $k$-jet approximation to the exact geodesic deviation equation of $\mathcal{S}$, linear on the deviation functions and invariant under an spec…

The pseudoconvex and disprisoning conditions for geodesics of linear connections are extended to the solution curves of general homogeneous sprays. The main result is that pseudoconvexity and disprisonment are jointly stable in the fine topology on the space of all homogeneous sprays of any degree of homogeneity.

In this work we show that for the geodesic spray $S$ of a Finsler function $F$ the most natural projective deformation $\widetilde{S}=S -2 λF\mathbb C$ leads to a non-Finsler metrizable spray, for almost every value of $λ\in \mathbb R$. This result shows how rigid is the metrizablility property with respect to certain …

The paper explores non-metrizability of projective deformations of Finsler sprays.

The projective metrizability problem can be formulated as follows: under what conditions the geodesics of a given spray coincide with the geodesics of some Finsler space, as oriented curves. In Theorem 3.8 we reformulate the projective metrizability problem for a spray in terms of a first-order partial differential ope…