# Problem Sheet F

This Problem Sheet is based on Lectures 10 and 11

- Problem 1 asks you identify the exponential map for $G = \mathrm{GL}(n)$. This is the reason for the name “exponential map”.
- Problem 2 discusses the relationship between left-invariant vector fields and right-invariant vector fields.
- Problem 3 asks you to complete the proof that $ \mathrm{ad}_v(w) = [v,w]$, which I lazily skipped in class.
- Problem 4 gives a condition that implies the tangent bundle $T M$ of a smooth manifold $M^n$ is diffeomorphic to $M \times \mathbb{R}^n$.
- Problem 5 asks you to apply Problem 4 when $M$ is a Lie group.
- Problem 6 is lots of fun. This proves 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 7 gives a nice way of constructing foliations.

Comments and questions?