• Problem 1 is shows differentiable homogeneous maps between vector spaces are automatically linear.
• Problem 2 is about building a parallel frame along a curve.
• Problems 3 and 4 explores how to create new connections from old (we will cover this more systematically later).
• Problem 5 is more on our favourite connection on $TS^m$ from the previous problem sheet.
• Problem 6 is about reducing a connection.
• Problem 7 shows how connections behave like tensor derivations.
• Problem 8 gives another way of thinking about flat connections. Remark: If you are not familiar with the universal cover, just skip this question.
• Problem 9 discusses left-invariant connections on Lie groups.