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