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 of an embedded submanifold in Euclidean space.