József Szilasi

On the Finsler-metrizabilities of spray manifolds

In this essentially selfcontained paper first we establish an intrinsic version and present a coordinate-free deduction of the so-called Rapcsák equations, which provide, in the form of second order PDE-s, necessary and sufficient conditions for a Finsler structure to be projectively related to a spray. From another viewpoint, the Rapcsák equations are the conditions for the Finsler-metrizability of a spray in a broad sense. Second, we give a reformulation in terms of 0-homogeneous Hilbert 1-forms of both this and another metrizability problem, called Finsler-metrizability in a natural sense. (The latter is just a Finslerian version of the classical inverse problem of the calculus of variations.) Finally, in our main theorem we provide a reduction of the Rapcsák equations to a first order PDE with an algebraic condition. The preparatory parts of the paper are devoted to a careful elaboration of the necessary technical tools, while in an Appendix the computational background is summarized.

Connections, sprays and Finsler structures

Modules, Algebras and Derivations Manifolds and Bundles Vector Fields, Tensors and Integration Structures on Tangent Bundles Sprays and Lagrangians Covariant Derivatives Theory of Ehresmann Connections Geometry of Spray Manifolds Finsler Norms and Finsler Functions.

Calculus along the tangent bundle projection and projective metrizability

We sketch an economical framework and a simple index-free cal- culus for direction-dependent objects, based almost exclusively on Berwald derivative. To illustrate the efficiency of these tools, we characterize projec- tively Finslerian sprays, and derive Hamels PDEs in an analytic version of Hilberts fourth problem.

A GENERALIZATION OF WEYL'S THEOREM ON PROJECTIVELY RELATED AFFINE CONNECTIONS

From a physical point of view, the geodesics in a four-dimensional Lorentzian spacetime are really significant only as point sets. In 1921 Weyl proved that two torsion-free covariant derivative operators DM and DM on a manifold M have the same geodesics with possibly different parametrizations if and only if there is a 1-form α on M such that D = D + α ⊗ 1 + 1 ⊗ α, where 1 is the identity (1,1) tensor on M. By a theorem of Ambrose, Palais and Singer [1], torsion-free covariant derivative operators are generated by affine sprays, and vice versa. More generally, any (not necessarily affine) spray induces a number of covariant derivatives in the tangent bundle τ of M, or in the pull-back bundle τ∗τ. We show that in the context of sprays, similarly to Weyls relation, a correspondence between the Yano derivatives can be detected.