The index of a geodesic in a Randers space and some remarks about the lack of regularity of the energy functional of a Finsler metric

In some recent papers, the relations existing between the metric properties of Randers spaces and the conformal geometry of stationary Lorentzian manifolds were discovered and investigated. In this note, we focus on the equality between the index of a geodesic in a Randers space and that of its lightlike lift in the associated conformal stationary spacetime. Moreover we make some remarks about regularity of the energy functional of a Finsler metric on the infinite dimensional manifold of $H^1$ curves between two points, in connection with infinite dimensional techniques in Morse Theory.Comment: Contribution to the proceedings of "Workshop on Finsler geometry and its applications", Debrecen, 24--29 May, 2009. 8 pages, AMSLaTex. v2 minor revision: typos fixed, references update

Harmonic Coordinates for the Nonlinear Finsler Laplacian and Some Regularity Results for Berwald Metrics

We prove existence of harmonic coordinates for the nonlinear Laplacian of a Finsler manifold and apply them in a proof of the Myers--Steenrod theorem for Finsler manifolds. Different from the Riemannian case, these coordinates are not suitable for studying optimal regularity of the fundamental tensor, nevertheless, we obtain some partial results in this direction when the Finsler metric is Berwald.Comment: AMS-LaTeX, 11 page

A variational setting for an indefinite Lagrangian with an affine Noether charge

We introduce a variational setting for the action functional of an autonomous and indefinite Lagrangian on a finite dimensional manifold. Our basic assumption is the existence of an infinitesimal symmetry whose Noether charge is the sum of a one-form and a function. Our setting includes different types of Lorentz-Finsler Lagrangians admitting a timelike Killing vector field.Comment: 42 pages, AMSLaTeX. v2: some small mistakes corrected in Examples 3.4,3.6, 3.9 and at the end of the proof of Prop. A

On the interplay between Lorentzian Causality and Finsler metrics of Randers type

We obtain some results in both Lorentz and Finsler geometries, by using a correspondence between the conformal structure (Causality) of standard stationary spacetimes on $M=\R\times S$ and Randers metrics on $S$. In particular, for stationary spacetimes, we give a simple characterization of when they are causally continuous or globally hyperbolic (including in the latter case, when $S$ is a Cauchy hypersurface), in terms of an associated Randers metric. Consequences for the computability of Cauchy developments are also derived. Causality suggests that the role of completeness in many results of Riemannian Geometry (geodesic connectedness by minimizing geodesics, Bonnet-Myers, Synge theorems) is played, in Finslerian Geometry, by the compactness of symmetrized closed balls. Moreover, under this condition we show that for any Randers metric there exists another Randers metric with the same pregeodesics and geodesically complete. Even more, results on the differentiability of Cauchy horizons in spacetimes yield consequences for the differentiability of the Randers distance to a subset, and vice versa.Comment: 26 pages, AMSLaTex. Accepted for publication on Rev. Mat. Iberoamericana. v2: improved presentation of the result