Journal of the
Korean Mathematical Society JKMS

This paper considers foundational issues related to connections in the tangent bundle of a manifold. The approach makes use of second order tangent vectors, i.e., vectors tangent to the tangent bundle. The resulting second order tangent bundle has certain properties, above and beyond those of a typical tangent bundle. In particular, it has a natural secondary vector bundle structure and a canonical involution that interchanges the two structures. The involution provides a nice way to understand the torsion of a connection.\par The latter parts of the paper deal with the Levi-Civita connection of a Riemannian manifold. The idea is to get at the connection by first finding its spray. This is a second order vector field that encodes the second order differential equation for geodesics. The paper also develops some machinery involving lifts of vector fields from a manifold to its tangent bundle and uses a variational approach to produce the Riemannian spray.

Keywords: connections in vector bundles, second order tangent vectors, anonical involution, torsion, geodesic spray, horizontal, canonical, and linear lifts, symmetric vector fields

MSC numbers: Primary 53B05; Secondary 53C05, 58A05, 58E10