arXiv subjects

Kazuhiro Shibuya

Publications and source records attributed to Kazuhiro Shibuya.

6 recordsLinked to original sources

arXiv2021

On a generalization of Monge-Ampère equations and Monge-Ampère systems

We discuss Monge-Ampère equations from the view point of differential geometry. It is known that a Monge-Ampère equation corresponds to a special exterior differential system on a 1-jet space. In this paper, we generalize Monge-Ampère equations and prove that a $(k+1)$st order generalized Monge-Ampère equation corresponds to a special exterior differential system on a $k$-jet space. Then its solution naturally corresponds to an integral manifold of the corresponding exterior differential system. Moreover, we verify that the Korteweg-de Vries (KdV) equation and the Cauchy-Riemann equations are examples of our equation.

math.DG↗

arXiv2018

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.

math.DG↗

arXiv2015

A variational problem for curves on Riemann-Finsler surfaces

We study the variational problem for $N$-parallel curves on a Finslerian surface by means of Exterior Differential Systems using Griffiths' method. We obtain the conditions when these curves are extremals of a length functional and write the explicit form of Euler-Lagrange equations for this type of variational problem.

math.DG↗

arXiv2014

Metric structures associated to Finsler metrics

We investigate the relation between weighted quasi-metric Spaces and Finsler Spaces. We show that the induced metric of a Randers space with reversible geodesics is a weighted quasi-metric space.

math.DG↗

arXiv2012

Rank two prolongations of second-order PDE and geometric singular solutions

In this present paper, we study geometric structures of rank two prolongations of implicit second-order partial differential equations (PDEs) for two independent and one dependent variables and characterize the type of these PDEs by the topology of fibers of the rank two prolongations. Moreover, by using properties of these prolongations, we give explicit expressions of geometric singular solutions of second-order PDEs from the point of view of contact geometry of second order.

math.DG↗

arXiv2012

Generalized Finsler structures on closed 3-manifolds

An (I,J,K)-generalized Finsler structure on a 3-manifold is a generalization of a Finslerian structure, introduced in order to separate and clarify the local and global aspects in Finsler geometry making use of the Cartan's method of exterior differential systems. In this paper, we show that there is a close relation between (I,J,1)-generalized Finsler structures and a class of contact circles, namely the so-called Cartan structures. This correspondence allows us to determine the topology of 3-manifolds that admit (I, J, 1)-generalized Finsler structures and to single out classes of (I, J, 1)-generalized Finsler structures induced by standard Cartan structures.

math.DG↗