Local Expression of Hessian Structures and Dissections on Manifolds

The Christoffel symbols are usually understood to be the coefficients of a connection or the components of a spray. In this note we show that they also arise naturally when one tries to obtain local characterizations of dissections of the bundle of second order tangent vectors on a manifold or of Hessian structures on a manifold. These results give an elementary explanation of known theorems relating connections, sprays, dissections and. Hessian structures. Further they show that the same local structure can be interpreted globally in widely different ways.