• Problem 1 is hell. Enjoy. 😈
• Problem 2 introduces a Leibniz operator, and asks you to prove they are local operators but not point operators.
• Problem 3 is a multilinear version of Proposition 20.25.
• Problem 4 asks you prove the general case of the Tensor Criterion.
• Problem 5 introduces the notion of a vertical bundle, which will be very useful when we discuss connections in Differential Geometry II.
• Problem 6 shows that the space of tensor derivations on a manifold can be identified with $\mathfrak{X}(M) \times \Gamma( \operatorname{End}(TM))$.
• Problem 7 looks horrendous, but it’s actually quite short. If you are scared, try proving the result first for $\mathcal{A}(A,B) = A \otimes B$.
• Problem 8 is about projective modules over commutative rings, and has nothing to do with geometry.
• Problem 9 shows that the space of sections of a vector bundle satisfies the hypotheses of Problem 8. This is used at the end of Lecture 21.