# Problem Sheet R

This Problem Sheet is based on Lectures 36 and 37.

- Problem 1 shows that the holonomy group of a Riemannian connection is a subroup of the orthogonal group.
- Problem 2 introduces the musical isomorphisms
^{[1]}between a Riemannian vector bundle and its dual. - Problem 3 asks you to prove that characterisic classes of odd degree invariant polynomials are always zero.
- Problem 4 asks you to verify that the Chern-Weil homomorphism is indeed a homomorphism.
- Problem 5 is the
*Whitney product formula*.*Remark:*The statement would be more complicated if one worked with cohomology with coefficients in $ \mathbb{Z}$. - Problem 6 asks you to compute the Pontryagin classes of $TS^n \to S^n$.

I didn’t make this name up! ↩︎