TITLE: A metric for Grassmannians
QUESTION [15 upvotes]: I'm reading an article by Ricardo Mañé, "The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces" (https://doi.org/10.1007/BF02585431). I'm having a technical problem. Sorry for my ignorance, but I would like a reference which explains how to equip the Grassmann manifolds with a metric.

REPLY [6 votes]: My "answer" is just a (still another) reformulation of Ian's.
Given two subspaces $V$, $W \subset \mathbb{R}^n$ with the same dimension, define their distance as:
$$
d(V,W):= \inf_J \| J - i_V\|
$$
where $i_V : V \to \mathbb{R}^n$ denotes the inclusion, $\|\mathord{\cdot}\|$ is the Euclidian operator norm, and $J$ runs over the linear maps $V \to \mathbb{R}^n$ whose such that $J(V) \subset W$. It's not too difficult to prove that the the infimum is attained at map $J=P_W|V$ (i.e. the orthogonal projection onto $W$ restricted to $V$). Trivially,
$$
\|P_W|V - i_V\| = \sup_{v\in V, \|v\|=1} \inf_{w\in W} \|w-v\|,
$$
so my formula is equivalent to Ian's.
As an easy consequence, we have the following interesting fact:

Fact: Let $T$ be a linear automorphism of $\mathbb{R}^n$. Then the induced trasformation of the grassmannian $G(k,n)$ is (bi-)Lipschitz with constant $\|T\|\,\|T^{-1}\|$. 

Proof:
$$
d(TV,TW) = \inf_{K(TV)\subset TW} \|K-i_{TV}\|=\inf_{J(V)\subset W}\big\|T(J-i_V)T^{-1}|_{TV}\big\| \le \|T\| \, \|T^{-1}\| \, d(V,W).
$$
(The proofs of the "fact" using other characterizations of $d$ become messier, I think.)