A systematic study of (smooth, strong) cone structures $${\mathcal {C}}$$ and Lorentz–Finsler metrics L is carried out. As a link between both notions, cone triples $$(\Omega ,T, F)$$, where $$\Omega $$ (resp. T) is a 1-form (resp. vector field) with $$\Omega (T)\equiv 1$$ and F, a Finsler metric on $$\ker (\Omega )$$, are introduced. Explicit descriptions of all the Finsler spacetimes are given, paying special attention to stationary and static ones, as well as to issues related to differentiability. In particular, cone structures $${\mathcal {C}}$$ are bijectively associated with classes of anisotropically conformal metrics L, and the notion of cone geodesic is introduced consistently with both structures. As a non-relativistic application, the time-dependent Zermelo navigation problem is posed rigorously, and its general solution is provided.

On the definition and examples of cones and Finsler spacetimes

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 operator P1 and a set of algebraic con- ditions on semi-basic 1-forms. We discuss the formal integrability of P1 using two sufficient conditions provided by Cartan{Kahler theorem. We prove in Theorem 4.2 that the symbol of P1 is involutive and hence one of the two conditions is always satisfied. While discussing the second condition, in Theorem 4.3 we prove that there is only one obstruction to the formal integrability of P1, and this obstruction is due to the curvature tensor of the in- duced nonlinear connection. When the curvature obstruction is satisfied, the projective metrizability problem reduces to the discussion of the algebraic conditions, which as we show are always satisfied in the analytic case. Based on these results, we recover all classes of sprays that are known to be projectively metrizable: flat sprays, isotropic sprays, and arbitrary sprays on 1- and 2-dimensional manifolds. We provide examples of sprays that are projectively metrizable without being Finsler metrizable.

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Projective Metrizability and Formal Integrability

In this work we show that for the geodesic spray S of a Finsler function F, the most natural projective deformation leads to a non-Finsler metrizable spray, for almost every value of λ ∈ ℝ. This result shows how rigid is the metrizablility property with respect to certain reparameterizations of the geodesics. As a consequence, we obtain that the projective class of an arbitrary spray contains infinitely many sprays that are not Finsler metrizable.

Projective and finsler metrizability: parameterization-rigidity of the geodesics

In this work we show that for the geodesic spray $S$ of a Finsler function $F$ the most natural projective deformation $\widetilde{S}=S -2 \lambda F\mathbb C$ leads to a non-Finsler metrizable spray, for almost every value of $\lambda \in \mathbb R$. This result shows how rigid is the metrizablility property with respect to certain reparameterizations of the geodesics. As a consequence we obtain that the projective class of an arbitrary spray contains infinitely many sprays that are not Finsler metrizable.

Projective and Finsler metrizability: parameterization-rigidity of the geodesics

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The inverse problem of the calculus of variations

To a system of second-order ordinary differential equations one can assign a canonical nonlinear connection that describes the geometry of the system. In this paper, we develop a geometric setting that also allows us to assign a canonical nonlinear connection to a system of higher-order ordinary differential equations (HODE). For this nonlinear connection we develop its geometry, and explicitly compute all curvature components of the corresponding Jacobi endomorphism. Using these curvature components we derive a Jacobi equation that describes the behavior of nearby geodesics to a HODE. We motivate the applicability of this nonlinear connection using examples from the equivalence problem, the inverse problem of the calculus of variations, and biharmonicity. For example, using components of the Jacobi endomorphism we express two Wuenschmann-type invariants that appear in the study of scalar third- or fourth-order ordinary differential equations.

A geometric setting for systems of ordinary differential equations

To a system of second order ordinary differential equations (SODE) one can assign a canonical nonlinear connection that describes the geometry of the system. In this work we develop a geometric setting that allows us to assign a canonical nonlinear connection also to a system of higher order ordinary differential equations (HODE). For this nonlinear connection we develop its geometry, and explicitly compute all curvature components of the corresponding Jacobi endomorphism. Using these curvature components we derive a Jacobi equation that describes the behavior of nearby geodesics to a HODE. We motivate the applicability of this nonlinear connection using examples from the equivalence problem, the inverse problem of the calculus of variations, and biharmonicity. For example, using components of the Jacobi endomorphism we express two Wuenschmann-type invariants that appear in the study of scalar third or fourth order ordinary differential equations.

The aim of this paper is to study from the point of view of linear connections the data with M a smooth (n+p)-dimensional real manifold, an n-dimensional manifold semi-Riemannian distribution on M, the conformal structure generated by g and W a Weyl substructure: a map such that W(ḡ) = W(g) - du, ḡ = eug;u ∈ C∞(M). Compatible linear connections are introduced as a natural extension of similar notions from Weyl geometry and such a connection is unique if a symmetry condition is imposed. In the foliated case the local expression of this unique connection is obtained. The notion of Vranceanu connection is introduced for a pair (Weyl structure, distribution) and it is computed for the tangent bundle of Finsler spaces, particularly Riemannian, choosing as distribution the vertical bundle of tangent bundle projection and as one-form the Cartan form.

https://www.math.uaic.ro/~mcrasm/depozit/IJGMMP.pdf

Oana Constantinescu

Papers

A geometric setting for systems of ordinary differential equations

A geometric setting for systems of ordinary differential equations

Sub-weyl geometry and its linear connections

A geometric settings for systems of ordinary differential equations

Helmholtz conditions and symmetries for the time dependent case of the inverse problem of the calculus of variations