Integrable geodesic flow in 3D and webs of maximal rank

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Abstract

We characterize geodesic flows, admitting two commuting quadratic integrals with common principal directions, in terms of the geodesic 4-webs such that the tangents to the web leaves are common zero directions of the integrals. We prove that, under some natural geometric hypothesis, the metric is of St\"ackel type.

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Sergey I. Agafonov (Sun,) studied this question.

synapsesocial.com/papers/68e75efdb6db6435876d608d https://doi.org/https://doi.org/10.48550/arxiv.2403.01459

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