Second-order Differential Equations, Exponential Maps, and Nonlinear Connections

The main purpose of this article is to introduce a comprehensive, unified theory of the geometry of all connections. We show that one can study any connection via a certain, closely associated second-order differential equation. One of the most important tools is our extended Ambrose-PalaisSinger correspondence. We extend the theory of geodesic sprays to arbitrary second-order differential equations, show that locally diffeomorphic exponential maps can be defined for any of them, and give a full theory of (possibly nonlinear) covariant derivatives for (possibly nonlinear) connections. In the process, we introduce vertically homogeneous connections. Unlike homogeneous connections, these complete our theory and allow us to include Finsler spaces among the applications. MSC(1991): Primary 53C15; Secondary 53C22, 58E10. −−−−−−−−−−−−−−−−−−−−−−−−−−→Υ⌣· ∞·←−−−−−−−−−−−−−−−−−−−−−−−−−−