The Finsler spaces of higher order (Q1818391)

Sixty-five years ago, A. Kawaguchi gave a first definition of the Finsler spaces of order \(k\geq 1\), denoted \(F^{(k)n}\), taking as fundamental function \(F(x,dx/dt,\dots,(d^kx/dt^k)/k!)\). He has preserved the known axioms of Finsler geometry, except that regarding the property of homogeneity. In a new axiom he asked that the arc length does not depend on the parametrization of the curve. In the case \(k>1\) the mentioned axiom leads to the Zermelo conditions. But the Kawaguchi's axiom is not convenient for a good definition of the Finsler spaces of order \(k>1\). A new definition of the higher order Finsler spaces was formulated by the author together with Sorin Sabau in the last year. A theorem of Euler type holds. Using this kind of homogeneity they could formulate the notion of higher order Finsler spaces. They study the geodesics, Cartan nonlinear connection and canonical metrical connection. Some examples are pointed out: the prolongation of the Riemannian or Finslerian spaces, the Randers spaces and the Kropina spaces of order \(k\). The applications of the geometry of spaces \(F^{(k)n}\) in mechanics or in theoretical physics are also remarkable.