A complete lift for semisprays

Details A complete lift for semisprays A complete lift for semisprays

2008-09-08

arxiv.org/abs/0809.1328v1

International Journal of Geometric Methods in Modern Physics, vol. 7, no 2 (2010), 267-287.

Mathematics

Differential Geometry

Scientific paper

10.1142/S0219887810004026

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 $S^c$ on $TM$. The motivation for this lift is two-fold: First, geodesics for $S^c$ correspond 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 $S^c$ uniquely determines $S$. We also study how symmetries and constants of motions for $S$ lift into symmetries and constants of motions for $S^c$.

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