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

# Author: dantegrevino

# 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 .

# Cover this

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

# Bounding the projective plane 2

Let be a natural number. Give an example of a compact -manifold such that .

# Rank isn’t like dimension

Let be the free group in the generators and . Consider the group morphism defined by . Prove that is a free group of infinite rank. (Hint: Think topologically!)

# Your degree is a chance for expansion

Let and be compact oriented smooth manifolds of the same dimension, with connected. Let be a smooth function. Take a regular value of and, for , define as if preserves orientation at and as if it reverses the orientation. We define the degree of as the number .

Prove the following extension theorem: Assume is a -dimensional sphere and is the boundary of a compact oriented -manifold . There exists a smooth extension of if and only if .