This Problem Sheet is based on Lecture 24, 25, and 26.

• Problem 1 shows that principal bundles behave nicely with respect to pullbacks.
• Problem 2 highlights a key difference between vector bundles and principal bundles.
• Problem 3 on the other hand shows that they are not all that different (compare this to Problem 4 on Problem Sheet H).
• Problem 4 is on homogeneous spaces (from Lecture 12).
• Problem 5 generalises the fact that left-invariant vector fields on a Lie group are in bijective correspondence with its Lie algebra.
• Problem 6 shows that the exterior differential is still skew-commutative when considered as an operator on vector-valued forms.
• Problem 7 is an example of where I claimed something was “trivial” in lecture and then realised afterwards that it was, in fact, not so trivial. But it’s still a good exercise. Enjoy! 😉