The geodesic flow on nilmanifolds associated to graphs

In this work we study the geodesic flow on nilmanifolds associated to graphs. We are interested in the construction of first integrals to show complete integrability on some compact quotients. Also examples of integrable geodesic flows and of non-integrable ones are shown.Comment: 22 page

Examples of naturally reductive pseudo-Riemannian Lie groups

We provide examples of naturally reductive pseudo-Riemannian spaces, in particular an example of a naturally reductive pseudo-Riemannian 2-step nilpotent Lie group $(N, _N)$, such that $_N$ is invariant under a left action and for which the center is degenerate. The metric does not correspond to a bi-invariant one.Comment: 7 pages, presented in XIX International Fall Workshop on Geometry and Physics, Porto (2010

The geodesic flow on nilmanifolds

In this paper we study the geodesic flow on nilmanifolds equipped with a left-invariant metric. We write the underlying definitions and find general formulas for the Poisson involution. As an example we develop the Heisenberg Lie group equipped with its canonical metric. We prove that a family of first integrals giving the complete integrability can be read off at the Lie algebra of the isometry group. We also explain the complete integrability on compact quotients and for any invariant metric.Comment: 24 page

Invariant metrics and Hamiltonian Systems

Via a non degenerate symmetric bilinear form we identify the coadjoint representation with a new representation and so we induce on the orbits a simplectic form. By considering Hamiltonian systems on the orbits we study some features of them and finally find commuting functions under the corresponding Lie-Poisson bracketComment: 16 pages corrected typos, changed contents (Prop. 3.4 and Theorem in Section 3