The geodesic flow on nilpotent Lie groups of steps two and three (Q2062708)

This work investigates the geodesic flow on \(k\)-step nilpotent Lie groups for \(k=2,3\). For such Lie groups in dimension at most five, the author obtains a left-invariant metric for which the geodesic flow is Liouville integrable. The main tools used to obtain Liouville integrability are Killing vector fields and symmetric Killing 2-tensors associated to quadratic invariant first integrals of the geodesic flow. The author obtains explicit conditions for a symmetric map to induce a first integral as well as conditions for invariant linear or quadratic polynomials to become first integrals. Several explicit involution formulas for first integrals are obtained. Moreover, they also investigate six-dimensional \(k\)-step nilpotent Lie groups for \(k=2,3\), and in most cases (all but two) show that these groups admit a left-invariant metric for which the geodesic flow is completely integrable.