Problem Sheet G

Published 9 months ago 1 min read

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.