Some remarks on the Finslerian version of Hilbert's fourth problem

The Finslerian version of Hilbert's fourth problem is the problem of finding projective Finsler functions. Alvarez Paiva (J. Diff. Geom. 69 (2005) 353-378) has shown that projective absolutely homogeneous Finsler functions correspond to symplectic structures on the space of oriented lines in R. with certain properties. I give new and direct proofs of his main results, and show how they are related to the more classical formulations of the problem due to Hamel and Rapcsak

On Landsberg spaces and the Landsberg-Berwald problem

This paper is concerned with the geometry of a class of Finsler spaces called Landsberg spaces. A Landsberg space may be characterized by the fact that its fundamental tensor is covariant constant along horizontal curves with respect to its Berwald connection. A Finsler space whose Berwald connection is affine is called a Berwald space. Berwald spaces are necessarily Landsbergian, but whether there are y-global Landsberg spaces which are not of Berwald type is not known. Resolving this question is the Landsberg-Berwald problem of the title. The paper deals with several topics in Landsberg geometry which are related mainly by the possibility that the results obtained may throw light on the Landsberg-Berwald problem. It is assumed throughout that the dimension of the base manifold is at least 3. It is shown that a Landsberg space over a compact base, which is R-quadratic, is necessarily Berwaldian. A model for the holonomy algebra of a Landsberg space is proposed. Finally, the technique of averaging the fundamental tensor over the indicatrix is discussed, and it is shown that for a Landsberg space, with the correct interpretations, the averaged Berwald connection is the Levi-Civita connection of the averaged metric

Finsler functions for two-dimensional sprays

I derive the general formula for a local Finsler function for any spray over a two-dimensional manifold specified by its geodesic curvature function relative to a given background Riemannian metric

Reduction and Reconstruction Aspects of Second-Order Dynamical Systems with Symmetry

We examine the reduction process of a system of second-order ordinary differential equations which is invariant under a Lie group action. With the aid of connection theory, we explain why the associated vector field decomposes in three parts and we show how the integral curves of the original system can be reconstructed from the reduced dynamics. An illustrative example confirms the results

Holonomy of a class of bundles with fibre metrics

This paper is concerned with the holonomy of a class of spaces which includes Landsberg spaces of Finsler geometry. The methods used are those of Lie groupoids and algebroids as developed by Mackenzie. We prove a version of the Ambrose-Singer Theorem for such spaces. The paper ends with a discussion of how the results may be extended to Finsler spaces and homogeneous nonlinear connections in general

Kahler and para-Kahler structures associated with finsler spaces of non-zero constant flag curvature

It was shown by R. L. Bryant (Houston J. Math. 28 (2002) 221262) that there is a canonical Kahler structure on the space of geodesics of a Finsler manifold whose flag curvature is constant and positive. A different construction is proposed in the present paper, leading instead to a Kahler structure on the slit tangent bundle of the Finsler space; it is based on the identification of an appropriate complex structure. The construction is easily adapted to apply to a Finsler space with constant negative flag curvature, when it gives a para-Kahler rather than a Kahler structure; the properties of this para-Kahler structure are explored

Generalized submersiveness of second-order ordinary differential equations

We generalize the notion of submersive second-order differential equations by relaxing the condition that the decoupling stems from the tangent lift of a basic distribution. It is shown that this leads to adapted coordinates in which a number of first-order equations decouple from the remaining second-order ones

Second-order dynamical systems of Lagrangian type with dissipation

We give a coordinate-independent version of the smallest set of necessary and sufficient conditions for a given system of second-order ordinary differential equations to be of Lagrangian form with additional dissipative forces