• 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 $TS^m \to S^m$.

  1. I didn’t make this name up! ↩︎

Comments and questions?