• 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?