TITLE: Canonical Metric on Grassmann Manifold
QUESTION [10 upvotes]: I was curious and quite clueless as to how we can equip the Grassmann Manifold with a canonical metric - I have yet to find anything upon this subject.

REPLY [7 votes]: In fact, $Gr(n,m)$ with its canonical metric induced from an Euclidean structure is one of the few spaces where you can write down solutions of the geodesic equations explicitly by a formula and write down a formula for the geodesic distance. See the following paper for this

MR1856419 Neretin, Yurii A. On Jordan angles and the triangle inequality in Grassmann manifolds. Geom. Dedicata 86 (2001), no. 1-3, 81–92.