# Problem Sheet F

- Problem 1 proves that the tangent bundle of a Lie group is trivial.
- Problem 2 asks you identify the exponential map for $G = \mathrm{GL}(m)$. This is the reason for the name “exponential map”.

- Problem 3 asks you to complete the proof that $ \mathrm{ad}_\xi(\zeta) = [\xi,\zeta]$, which I lazily skipped in class.
- Problem 4 gives an alternative characterisation of proper actions.
- Problem 5 asks you to prove that left/right translation is a proper action.
- Problem 6 shows that the orbits are always closed subsets.
- Problem 7 shows that closed integral manifolds are automatically maximal.
- Problem 8 gives a nice way of constructing foliations.
- Problem 9 asks you to prove that any topological space which is simultaneously a topological group and a topological manifold can admit at most diffeomorphism class of smooth structures that turns it into a Lie group.
- Problem 10 shows that second countability is not needed in the definition for connected Lie groups.
- Problem 11 is about the group of automorphisms of a Lie group.

Comments and questions?