Lorentz-Finsler geometry reveals new and old inequalities.
problem Finding new inequalities using Lorentz-Finsler geometry.
method Applying reverse Cauchy-Schwarz and reverse triangle inequalities in Lorentz-Finsler geometry.
result Proved new and refined inequalities, including refinements of Aczél's inequality.
A new model uses Lorentz-Finsler geometry to predict wave propagation.
problem Modeling wave propagation in anisotropic and rheonomic media.
method Identifying wave trajectories as lightlike pregeodesics of a specific Lorentz-Finsler metric, solving ODE systems.
result Wave trajectories can be easily computed in real time.
New theorem splits weighted Lorentz-Finsler manifolds into simpler parts.
problem Understanding the geometry of weighted Lorentz-Finsler manifolds.
method Developed a splitting theorem using weighted Berwald spacetimes and Busemann functions.
result Weighted Lorentz-Finsler manifolds with certain properties split into simpler isometric translations.
We develop the theory of weighted Ricci curvature in a weighted Lorentz-Finsler framework and extend the classical singularity theorems of general relativity. In order to reach this result, we generalize the Jacobi, Riccati and Raychaudhuri equations to weighted Finsler spacetimes and study their implications for the e…
Navigation in Lorentz Finsler geometry induces isoparametric hypersurfaces.
problem Defining and analyzing isoparametric hypersurfaces in Lorentz Finsler geometry.
method Using a navigation process with a Finsler metric and a tangent vector field, isoparametric functions and hypersurfaces are defined and analyzed.
result Local correspondences between isoparametric functions and hypersurfaces are established.
Unified geometry for relativity and beyond.
problem Unified geometric framework for relativity and beyond.
method Unified Lorentz-Finsler geometry.
result Unified framework for relativity and beyond.
The optimal transport problem is studied in the context of Lorentz-Finsler geometry. For globally hyperbolic Lorentz-Finsler spacetimes the first Kantorovich problem and the Monge problem are solved. Further the intermediate regularity of the transport paths is studied. These results generalize parts of Bertrand & Puel…
Introduces a variational framework for indefinite Lagrangians with specific symmetries.
problem Handling indefinite Lagrangians with complex symmetries.
method Develops a variational setting for an indefinite Lagrangian with a specific Noether charge.
result Validates the existence of a variational setting for a broad class of Lagrangians.
Revisits Finsler spacetimes from inertial observer perspective.
problem Physical foundations of relativistic spacetimes.
method Inertial observers and double linear approximation.
result Finsler spacetimes are defined by dropping the second linearization.
Wave propagation framework using cone structures and observers' vector fields.
problem Describing classic wave propagation in anisotropic media.
method Introduces a cone structure C and an observers' vector field ∂t to describe wave propagation. result Reduces the PDE for wavefronts to ODE for cone geodesics of C. A new model predicts wildfire spread with wind and slope effects.
problem Predicting wildfire spread with wind and slope effects.
method Geometric model based on Lorentz-Finsler framework, considering wind and slope.
result Infinitesimal wavefronts are no longer restricted to be elliptical, allowing for more accurate predictions.
Generalizes Fermat's principle for wave propagation in cone structures.
problem Wave propagation in complex media with discontinuities and anisotropy.
method Generalizes Fermat's principle to smooth interfaces separating two cone structures representing wave propagation in various media.
result Conditions for critical points of arrival time functional, generalizing Snell's law and reflection.
In a recent work I showed that the family of smooth steep time functions can be used to recover the order, the topology and the (Lorentz-Finsler) distance of spacetime. In this work I present the main ideas entering the proof of the (smooth) distance formula, particularly the product trick which converts metric stateme…
Researchers developed volume comparison theorems in Finsler spacetimes.
problem Volume comparison in Finsler spacetimes with specific curvature conditions.
method Riccati equation techniques applied to (1+n)-dimensional Lorentz--Finsler manifolds. result Established volume comparison theorems for standard sets in Lorentzian volumes (SCLVs).
The dual problem of optimal transportation in Lorentz-Finsler geometry is studied. It is shown that in general no solution exists even in the presence of an optimal coupling. Under natural assumptions dual solutions are established. It is further shown that the existence of a dual solution implies that the optimal tran…
We study Finsler spacetimes and Killing vector fields taking care of the fact that the generalized metric tensor associated to the Lorentz-Finsler function L is in general well defined only on a subset of the slit tangent bundle. We then introduce a new class of Finsler spacetimes endowed with a timelike Killing vect…
The paper characterizes global hyperbolicity in Lorentzian manifolds without relying on manifold topology.
problem Characterizing global hyperbolicity in smooth Lorentzian manifolds without assuming manifold topology.
method Two formulations of global hyperbolicity: one using chronological diamonds and the other using properties of the Lorentzian distance function.
result The second formulation is equivalent to the definition of `Lorentzian metric space' and introduces the concept of d-reflectivity. We adopt a vierbein formalism to study pseudo-Finsler spaces modeled on a pseudo-Minkowski space. We show that it is possible to obtain closed expressions for most of the geometric objects of the theory, including Berwald's curvature, Landsberg's tensor, Douglas' curvature, non-linear connection and Ricci scalar. These…
A systematic study of (smooth, strong) cone structures $\C$ and Lorentz-Finsler metrics L is carried out. As a link between both notions, cone triples (Ω,T,F), where Ω (resp. T) is a 1-form (resp. vector field) with Ω(T)≡1 and F, a Finsler metric on ker(Ω), are introduced. Explicit descriptions o…
Optimizes transport in Finsler spacetimes with lower Ricci bounds.
problem Optimizing transport in Finsler spacetimes with lower Ricci bounds.
method Using optimal transport and weighted Ricci curvature bounds.
result Proves timelike curvature-dimension condition for Finsler spacetimes.
New calculus on spacetimes for nonlinear differential equations.
problem Nonlinear differential equations on metric measure spacetimes.
method Introduces maximal weak subslope and variational calculus.
result Establishes a comparison theorem for nonlinear p-d'Alembertian. Revisits stress-energy tensor in Finsler spacetimes, showing it's anisotropic.
problem Defining stress-energy tensor in Finsler spacetimes.
method Uses both heuristic and Lagrangian approaches, revisits divergence and conservation laws.
result Introduces a natural anisotropic Lie bracket derivation leading to the Chern anisotropic connection.
New algorithms solve time-dependent navigation problems, including zig-zag paths.
problem Finding optimal paths in moving water with varying speed profiles.
method Lorentz-Finsler geometry, novel computational algorithms.
result Optimal paths may involve zig-zag trajectories due to tacking behavior.