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arXiv · 1112.0228

k-Parameter geodesic variations

Ioan Bucataru·Matias F. Dahl
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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.

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Ioan Bucataru, Matias F. Dahl. 2011-12-01. k-Parameter geodesic variations. https://doi.org/10.1016/j.geomphys.2012.06.006

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