Metric equivalence of path spaces (Q2763763)

\textit{J. Douglas} [The general geometry of paths, Ann. Math. (2) 29, 143-168 (1928; JFM 54.0757.06)] formulated a curved version of projective geometry, which has paths given by the solutions of a system of second order differential equations and distance measured by a Finsler function. In general the differential equations do not describe geodesics of the Finsler structure. Douglas called this situation the `metric geometry' of a path space. Now for this geometry the nonlinear connection is constructed together with the first order invariant (torsion) tensors and the second order invariant (curvature) tensors. It is shown that two pairs of a system of ordinary differential equations and a metric are locally equivalent if and only if their invariants and covariant derivative thereof have the same rank and order and coincide on a neighborhood in classifying space. Some comments are made about applications in biological models.