kk-parameter geodesic variations

Suppose SS is a semispray on a manifold MM. We know that the complete lift ScSc of SS is a semispray on TMTM with the property that geodesics of ScSc correspond to Jacobi fields of SS. In this note we generalise this result and show how geodesic variations of kk variables are related to geodesics of the kkth iterated complete lift of SS. Moreover, for sprays (that is, homogeneous semisprays) we show how geodesic variations of (n−1)(n−1) variables are related to a natural generalisation of Jacobi tensors.