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Are there many examples of completely integrable geodesic flows (in the sense of Liouville), with say n integrals $f_1,\cdots, f_n$ such that everywhere, the differentials $(df_1,\cdots,df_n)$ are linearly independant ?

(recall that in the usual definition of completely integrable flows, one only requires that these are independent almost everywhere or in a dense open set).

Thanks !

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  • $\begingroup$ A trivial example: geodesic flow on Euclidean $n$-space is straight-line motion, and the momenta $p_1, \dots, p_n$ are first integrals with obviously independent differentials. $\endgroup$ – Hans Lundmark Oct 22 '13 at 21:09
  • $\begingroup$ Yes of course you're right, but what about a geodesic flow on a more complicated Riemannian manifold ? $\endgroup$ – user102589 Oct 23 '13 at 8:25
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I cannot comment; thus, I am posting this as an answer.

It is really strong to ask for global action-angle coordinates (see Duistermaat's article on that matter for a general discussion). However, there is a whole class of examples of nontrivial integrable geodesic flows on Lie groups: look for "Euler equations of finite-dimensional Lie groups" by Mischenko and Fomenko.

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