Where can I find the most direct and simplest presentation of what geodesics on a (complex) Grassmannian look like? I know how to do it from scratch, but, if I want to provide a reference to, say, a graduate student in EE who doesn't want to deal with any unnecessary abstract mathematical machinery, what should I point him to?



Geodesics on a Grassmannian
Grassmanians are symmetric spaces, and symmetric spaces are "geodesic orbit spaces", that is, their geodesics are orbits of their group of isometries. Your Grassmanians, in particular, are of the form $SU(p+q)/SU(p)\times SU(q)$. If $g$ is the Lie algebra of the big group and $h\subseteq g$ the Lie algebra of the subgroup, then there is a $SU(p)\times SU(q)$-invariant complement $p$ to $h$ in $g$. The geodesics are the orbits of the $1$-parameter subgroups of $SU(p+q)$ whose tangent vectors are in $p$.


So to compute the geodesics, you need only find that complement $p$ and compute exponentials...


