On the integrability of geodesic flows of submersion metrics (Q1863693)

Let \((Q,g)\) be a compact Riemannian manifold with a completely integrable geodesic flow, \(G\) a compact connected Lie group acting freely on \(Q\) by isometry. In this paper, complete integrability of geodesic flow on \(Q/G\) equipped with the submersion metric is proved under a natural assumption, by using the Mishchenko-Fomenko-Nekhoroshev theorem on noncommutative integration of Hamiltonian systems [\textit{A. S. Mishchenko} and \textit{A. T. Fomenko}, Funct. Anal. Appl. 12, 113-121 (1978); translation from Funkt. Anal. Prilozh. 12, No. 29, 46-56 (1978; Zbl 0396.58003)]. To show this, first the concept of noncommutative integrability is reviewed. Let \(M\) be a \(2n\)-dimensional symplectic manifold, \(({\mathcal F},\{\cdot,\cdot\}\) a Poisson subalgebra of \((C^\infty (M),\{\cdot,\cdot\})\) and let \(F_x\) be the subspace of \(T_x^*M\) generated by \(df(x)\), \(f\in{\mathcal F}\). If, \(\dim F_x=l\), and \(\dim\text{ker} \{\cdot, \cdot\} |_{F_x}=r\) on an open dense set of \(M\), then \(l\) and \(r\) are denoted by \(d\dim {\mathcal F}\) and \(d\text{ind}{\mathcal F}\), respectively. \({\mathcal F}\) is said to be complete if \(d\dim {\mathcal F}+dd \text{ind} {\mathcal F}=2n\). The Hamiltonian system \(\dot x=\text{sgrad} H(x)\) on \(M\) is called completely integrable in the noncommutative sense, if it possesses a complete algebra of first integrals \({\mathcal F}\). If the geodesic flow of \((Q,g)\) is completely integrable by means of a complete algebra \({\mathcal F}\) of first integrals, then the phase space \(T^*Q\) is foliated by invariant \(d\text{ind} {\mathcal F}\)-dimensional isotropic tori in an open dense set \(\text{reg} T^*Q\). Then it is proved that the geodesic flow on \(Q/G\) endowed with the submersion metric is completely integrable if \(\text{reg} T^*Q\) intersects with the space of horizontal vectors in a dense set. As an example, metrics on \(G\times G\) with combi-quotient \(T/U\), where \(U\) is any subgroup of \(T\times T\), \(T\) the maximal torus of \(G\), are constructed in Example 10.