# Problem Sheet I

- Problem 1 is hell. Enjoy. ðŸ˜ˆ
- 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 itâ€™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.

Comments and questions?