# 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 $-1$.

# Bend over

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

# Lines in flags

Let $F_1 \subseteq \dots \subseteq F_n$ and $F'_1 \subseteq \dots \subseteq F'_n$ be complete flags over an $n$-dimensional vector space $V$. Show that there is a set of $n$ lines $L_1,\dots, L_n$ compatible with the two flags, in the sense that there exists a permutation $\sigma \in S_n$ such that $F_i = L_1 \oplus \dots \oplus L_i$ and $F'_i = L_{\sigma(1)}\oplus \dots \oplus L_{\sigma(i)}$.

# Cover this

Let $f:M\to N$ be a local isometry between connected riemannian manifolds. If $M$ is complete, then $N$ is complete and $f$ is a covering map.

# Kill the Killing form

Let $\mathfrak g$ be a Lie algebra and $K$ its Killing form. Is the kernel of $K$ always equal to the radical of $\mathfrak g$?

# Centers

Let $G$ be a connected Lie subgroup of $\mathrm{GL}_n$. Then $A\in G$ commutes with all of $G$ iff it commutes with all of $\mathfrak{g}$.