# Problem Sheet C

This Problem Sheet is based on Lectures 4 and 5.

- Problem 1 is asks you to prove the cotangent bundle is a smooth manifold.
*Hint:*Mimic the proof that the tangent bundle was a smooth manifold. - Problem 2 asks you to prove that the derivative of a map is smooth as a map betweem tangent bundles.
- Problem 3 discusses the
*fibrewise derivative*. - Problems 4 asks you to show that injective immersions are the same as embeddings when the domain manifold is closed.
- Problem 5 discussed the tangent space to a product manifold $M \times N$.
- Problem 6 shows that the graph of a smooth function $f \colon O \subset \mathbb{R}^n \to \mathbb{R}$ (where $O$ is open in $ \mathbb{R}^n$) is always a smooth embedded $n$-dimensional submanifold of $ \mathbb{R}^{n+1}$.
- Problem 7 allows you to finally identify the tangent space to the sphere with what it “ought” to be.
- Problem 8 introduces the
*normal bundle*

Comments and questions?