§ The grassmanian, handwavily
The grassmanian is a manifold consisting of, roughly, dimensional subspaces
of an dimensional vector space.
Here, I'll record derivations of how we represent grassmanians, the exponential
map, logarithm map, and the parallel transport with "physicist style"
reasoning. Really, one needs to be careful because the grassmanian is a
quotient of the non-compact steifel manifold, so we need to be careful about
choosing representatives and whatnot. However, nothing beats physicist
reasoning for intuition, so I'm going to do all the derivations in that style.