• Problem 1 shows that projective spaces are homogeneous spaces.
• Problem 2 is a partial "converse" to the Quotient Manifold Theorem.
• Problem 3 exhibits a key difference between vector and principal bundles.
• Problem 4 is similar to the Fibre Bundle Construction Theorem — most of the work is in formulating the correct hypotheses. After that the result practically proves itself.
• Problem 5 shows that the structure group of a vector bundle can always be reduced to the orthogonal group (hint: Gram Schmidt).
• Problems 6 shows there are only two rank 1 bundles over $S^1$.
• Problem 7 gives an example of an $S^1$ fibre bundle which is not a principal $S^1$-bundle.
• Problem 8 is fun. (You might need a hint for this one.)
• Problem 9 shows that in the definition of a homogeneous space we may assume the action is effective.
• Problem 10 is about principal subbundles. Principal subbundles will play no role in Differential Geometry I. However they will be important in Differential Geometry II.
• Problem 11 was meant to be on the last sheet but I forgot it.