31 research outputs found
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 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 and Randers metrics on . 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 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