Some structural considerations on the theory of fields in higher order spaces (Q1369029)

A Kawaguchi space \(K^{(m)n}\) over a real, \(n\)-dimensional manifold \(M\) is given by a positive function \(F\) defined on the bundle of accelerations of order \(m\) such that the integral \(s= \int^1_0 F(x, {dx \over dt}, \dots, {d^mx \over dt^m}) dt\) does not depend on the parametrization of the curve \(c\): \(t\in [0,1] \to(x^i(t)) \in M\). The author discusses two problems: (1) What kind of metric tensor must to be adopted for \(K^{(m) n}\), which is convenient for a theory of physical fields? (2) What kind of nonlinear connection, depending on \(F\), is convenient for a spin gauge field theory?