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 $TS^n \to S^n$.

1. I didn’t make this name up! ↩︎