k-Parameter geodesic variations

TL;DR

This paper generalizes the relationship between geodesic variations and iterated complete lifts of a semispray, extending classical results to k-parameter variations and introducing a connection to Jacobi tensors for sprays.

Contribution

It introduces a framework linking k-parameter geodesic variations to the geodesics of iterated complete lifts, generalizing known results for Jacobi fields.

Findings

Generalization of geodesic-Jacobi correspondence to k-parameter variations.

Establishment of a relation between geodesic variations and iterated complete lifts.

Extension to sprays and the connection to Jacobi tensors.

Abstract

Suppose $S$ is a semispray on a manifold $M$. We know that the complete lift $S^c$ of $S$ is a semispray on $TM$ with the property that geodesics of $S^c$ correspond to Jacobi fields of $S$. In this note we generalize this result and show how geodesic variations of $k$-variables are related to geodesics of the $k$th iterated complete lift of $S$. Moreover, for sprays (that is, homogeneous semisprays) we show how geodesic variations of $(n-1)$-variables are related to a natural generalisation of Jacobi tensors.