# Problem Sheet G

- 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.

Comments and questions?