Kazuhiro Shibuya

Metric structures associated to Finsler metrics

AbstractWe investigate the relation between weighted quasi-metric Spaces and FinslerSpaces. We show that the induced metric of a Randers space with reversiblegeodesics is a weighted quasi-metric space. 1 Introductionand Motivation Riemannian spaces can be represented as metric spaces. Indeed, for a Riemannian space(M,a) we can define the induced metric space (M,d α ), with the metricd α : M× M→ [0,∞), d α (x,y) := inf γ∈Γ xy Z ba α(γ(t),γ˙(t))dt, (1.1)where Γ xy := {γ: [a,b] → M| γ(piecewise) C ∞ -curve,γ(a) = x,γ(b) = y} is the set ofcurves joining points xand y, ˙γ(t) := dγ(t)dt the tangent vector to γat γ(t), and α(x,X)the Riemannian norm of the vector X∈ T x M. It is easy to see that d α is a metric on M,i.e. it satisfies the axioms:1. Positiveness: d α (x,y) >0 if x6= y, d α (x,x) = 0,2. Symmetry: d α (x,y) = d α (y,x),3. Triangle inequality: d α (x,y) ≤ d α (x,z) +d α (z,y),for any x,y,z∈ M.More general structures than Riemannian ones are Finsler structures (see [BCS00],[S01], [MHSS01] for definitions).Similarly with the Riemannian case, one can define the induced metric of a Finslerspace (M,F) byd

MOVING FRAMES ON GENERALIZED FINSLER STRUCTURES

We study the relation between an R-Cartan structure {\alpha} and an (I, J, K)- generalized Finsler structure on a 3-manifold showing the difficulty in finding a general transformation that maps these structures each other. In some particular cases, the mapping can be uniquely determined by geometrical conditions.

Geodesics on strong Kropina manifolds

We study the behavior of the geodesics of strong Kropina spaces. The global and local aspects of geodesics theory are discussed. Our theory is illustrated with several examples.

ON IMPLICIT SECOND-ORDER PDE OF A SCALAR FUNCTION ON A PLANE VIA DIFFERENTIAL SYSTEMS

In the present paper, we study implicit second-order PDEs (i.e. partial differential equations) of single type for one unknown function of two variables. In particular, by using the theory of differential systems, we give a geometric characterization of PDEs which have a certain singularity. Moreover, we provide a new invariant of PDEs under contact transformations.