# Problem Sheet C

- Problem 1 discussed the tangent space to a product manifold $M \times N$ (pay attention to the word “canonical”).
- Problem 2 asks you to prove that the derivative of a map is smooth as a map betweem tangent bundles.
- Problems 3 asks you to show that injective immersions are the same as embeddings when the domain manifold is closed.
- Problem 4 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 5 allows you to finally identify the tangent space to the sphere with what it “ought” to be.
- Problem 6 introduces the
*normal bundle*of an embedded submanifold in Euclidean space. - Problem 7 shows that constant rank bijections are automatically diffeomorphisms.
- Problem 8 introduces the notion of a
**tranversal intersection**and generalises the Implicit Function Theorem. - Problem 9 shows that covering spaces of smooth manifolds are automatically smooth manifolds.

Comments and questions?