Integrable geodesic flows on \(n\)-step nilmanifolds
The author considers the following question: ``A Riemannian metric on a smooth manifold induces a geodesic flow \(\varphi_t\) on \(T^*M^n\), a \(\dots\) Hamiltonian flow. It is very rare for this flow to be Liouville integrable, \(\dots\) what are the topological implications of Liouville integrability?'' The main theorem in this paper, is: (Integrability of the geodesic flow) Let \(n \geq 2\) and \(N \in gl(n,\mathbb{R})\) be nilpotent: \(N^k = 0\) for some \(k \geq 0\). Define the multiplication \(\star\) on \(g = G_N = \mathbb{R} \times \mathbb{R}^n\) by \[ (x,y) \star (x^\prime,y') : = (x + x^\prime, y + \exp (xN) y'), \] where exp is the usual matrix exponential function. Let \(D \leq G\) be a lattice and \(g\) a left-invariant metric on \(G\), then the geodesic flow of \(H(p,q) = \frac 12 g^{-1} (p,p)\) is Liouville integrable on \(T^\star (D|G) \) with real analytic first integrals and a single \(C^\infty\) first integral. The author has previously worked in explicit examples of integrable geodesic flows [New examples of integrable geodesic flows, Asian J. Math. 4, No. 3, 515-526 (2000; Zbl 1116.37312)]. For other works in the same area, see \textit{V. S. Matveev} and \textit{P. J. Topalov}, Geodesic equivalence of metrics as a particular case of integrability of geodesic flows, Theor. Math. Phys. 123, No. 2, 651-658 (2000); translation from Teor. Mat. Fiz. 123, 285-293 (2000; Zbl 0996.53054).
scientific article; zbMATH DE number 1563135
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