# Just one model

Two riemannian $n$-manifolds of sectional curvature constantly $k\in\mathbb{R}$ 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 $-1$.

# 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.

# Bounding the projective plane 2

Let $n$ be a natural number. Give an example of a compact $2n$-manifold $M$ such that $\partial M = \mathbb{R}P^{2n-1}$.

# Rank isn’t like dimension

Let $F_2(a,b)$ be the free group in the generators $a$ and $b$. Consider the group morphism $f: F_2(a,b) \to \mathbb{Z}$ defined by $f(a)=f(b)=1$. Prove that $Ker(f)$ is a free group of infinite rank. (Hint: Think topologically!)

# Your degree is a chance for expansion

Let $X$ and $Y$ be compact oriented smooth manifolds of the same dimension, with $Y$ connected. Let $f : X \to Y$ be a smooth function. Take $y \in Y$ a regular value of $f$ and, for $x \in f^{-1}(y)$, define $o(x)$ as $+1$ if $f$ preserves orientation at $x$ and as $-1$ if it reverses the orientation. We define the degree of $f$ as the number $d(f) = \sum_{x\in f^{-1}(y)}o(x)$.

Prove the following extension theorem: Assume $Y=S^k$ is a $k$-dimensional sphere and $X=\partial W$ is the boundary of a compact oriented $(k+1)$-manifold $W$. There exists a smooth extension $F:W\to Y$ of $f$ if and only if $d(f)=0$.