General connections, exponential maps, and second-order differential equations (Q2901313)

The authors study general (non-linear) connections on the tangent bundle \(TM\) of a smooth manifold \(M\), which are defined via a distribution complementary to the vertical subbundle. Several subclasses of general connections containing the linear connections are defined. Next, general connections are related to (quasi)sprays and special second order differential equations via their geodesics, with the subclasses having special properties. Geodesics also lead to various versions of exponential maps. Also, an operation of covariant derivative can be constructed for a general connection, and the authors develop a notion of torsion freeness valid in this generality. The case of connections coming from Finsler structures is discussed in more detail. Finally, some results on geodesic connectivity and stability are proved.