Two riemannian -manifolds of sectional curvature constantly are locally equivalent.
Give an example of a hyperbolic manifold with finite volume and non hyperbolic fundamental group.
Remark: It has to be non compact.
Prove that a compact oriented surface of genus greater or equal to two has a hyperbolic structure, i.e. a metric with gaussian curvature constantly .
Prove that the torus cannot have a metric with always negative curvature.
Let and be complete flags over an -dimensional vector space . Show that there is a set of lines compatible with the two flags, in the sense that there exists a permutation such that and .
Let be a local isometry between connected riemannian manifolds. If is complete, then is complete and is a covering map.
Let be a Lie algebra and its Killing form. Is the kernel of always equal to the radical of ?