# Problem Sheet N

- Problem 1 studies the space of connections, and proves that this is an affine space modelled on $ \Omega^1(M, \operatorname{End}(E))$.
- Problem 2 relates $d^{ \nabla}$ and $d^{ \nabla^{ \operatorname{End}}}$.
- Problem 3 introduces the idea of a $G$-connection.
- Problem 4 shows that the holonomy group of a Riemannian connection is a subroup of the orthogonal group.
- Problem 5 introduces the
*musical isomorphisms*^{[1]}between a Riemannian vector bundle and its dual. - Problem 6 asks you to prove that characterisic classes of odd degree invariant polynomials are always zero.
- Problem 7 asks you to verify that the Chern-Weil homomorphism is indeed a homomorphism.
- Problem 8 is the
*Whitney product formula*.*Remark:*The statement would be more complicated if one worked with cohomology with coefficients in $\mathbb{Z}$. - Problem 9 asks you to compute the Pontryagin classes of $TS^m \to S^m$.

I didn’t make this name up! ↩︎