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As far as I understand, there isn't really a sense in which the integrator is noneuclidean. "Isokinetic" was coined by https://pubs.aip.org/aip/jcp/article-abstract/118/6/2510/438540/Algorithms-and-novel-applications-based-on-the, which we discovered is the first paper proposing the form of this integrator (but without the stochasticity introduced in the MCHMC papers)