Two riemannian -manifolds of sectional curvature constantly are locally equivalent.

# Not so hyperbolic

Give an example of a hyperbolic manifold with finite volume and non hyperbolic fundamental group.

Remark: It has to be non compact.

# Bend over 2

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 .

# Bend over

Prove that the torus cannot have a metric with always negative curvature.

# Lines in flags

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 .

# Cover this

Let be a local isometry between connected riemannian manifolds. If is complete, then is complete and is a covering map.

# Kill the Killing form

Let be a Lie algebra and its Killing form. Is the kernel of always equal to the radical of ?