Local midpoints on smooth manifolds

In this paper we consider three methods for obtaining midpoints, primarily midpoints of geodesics of sprays, but also midpoints of symmetry (in symmetric spaces), and metric midpoints (in Riemannian manifolds). We derive general conditions under which these approaches yield the same result. We also derive a version of the Lie–Trotter formula based on the midpoint operation and use it to show that continuous maps preserving (local) midpoints are smooth.