Problem Sheet K

Published 4 months ago 1 min read

Problem 1 and 2 are about singular cubes.
Problem 3 is a nice statement about symplectic manifolds.
Problem 4 asks you to find a closed non-exact form on $ \mathbb{R}^m \setminus \{0\}$.
Problem 5 is used in Lecture 27.
Problem 6 investigates how diffeomorphisms affect integrals.
Problem 7 shows that on Lie groups we can also integrate functions.
Problem 8 is lots of fun. This introduces the degree of a smooth map between compact oriented connected smooth manifolds, and then uses this to prove the infamous Hairy Ball Theorem.