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 .
Let be a local isometry between connected riemannian manifolds. If is complete, then is complete and is a covering map.
A plane flies 100 km north, 100 km east, 100 km south and then 100 km west. Give a necessary and sufficient condition for it to land on the same spot where it took off.
Any compact connected Lie group has Euler characteristic zero. What are the possible values for the Euler characteristic of a connected, but not necessarily compact Lie group?
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 .