A COMPLETE LIFT FOR SEMISPRAYS

Abstract

In this paper, we define a complete lift for semisprays. If S is a semispray on a manifold M, its complete lift is a new semispray Scon TM. The motivation for this lift is two-fold: First, geodesics for Sccorrespond to the Jacobi fields for S, and second, this complete lift generalizes and unifies previously known complete lifts for Riemannian metrics, affine connections, and regular Lagrangians. When S is a spray, we prove that the projective geometry of Scuniquely determines S. We also study how symmetries and constants of motions for S lift into symmetries and constants of motions for Sc.