This Problem Sheet is based on Lectures 45, 46, and 47.

• Problem 1 intrduces Killing fields on a Riemannian manifold $(M,m)$, which are vector fields $X$ such that $\mathcal{L}_X(m) = 0$.
• Problem 2 asks you to show that if $\varphi \colon M \to N$ is an isometric map then for $w \in T_{ \varphi(x)}N$, the tangential component $w^\top$ is simply the orthogonal projection of $w$ onto $D \varphi(x)[T_xM]$. (This is a rather simpler definition than the one I gave in class!)
• Problem 3 is about Riemannian coverings, and requires you to know a little bit of covering space theory in order to solve it.
• Problem 4 is an example of a Riemannian submersion. We won’t define these in generality in this course as we won’t need them—roughly speaking they are the dual concept to an isometric map (which is necessarily an immersion), and is the natural class of maps $\varphi \colon M \to N$ to look at in the Riemannian category when $\dim M \ge \dim N$.
• Problem 5 completes our disucssion of holonomy groups.