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

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 a connection via a certain, closely associated second-order differential equation. One of the most important results is our extended Ambrose-Palais-Singer correspondence. We extend the theory of geodesic sprays to certain second-order differential equations, show that locally diffeomorphic exponential maps can be defined for all, 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 in a completely consistent manner. MSC(2000): Primary 53C05; Secondary 53C15, 53C22. −−−−−−−−−−−−−−−−−−−−−−−−−−→Υ⌣· ∞·←−−−−−−−−−−−−−−−−−−−−−−−−−−