Given the set of $k$ dimensional subspaces of $\mathbb{R}^n$, we can define a metric on that set by $$d(L_1,L_2) = \sup_{|x| = 1} | P_{L_1}(x) - P_{L_2}(x)|$$
where $P_{L}(x)$ is the orthogonal projection of $x$ onto $L^\perp$, i.e. the "distance" from the point $x$ to the subspace $L$.
Lets take the simple case of $n=2$, $k=1$, so we have the set of lines through the origin in $\mathbb{R}^2$ (the projective space), is there a geometric interpretation for the metric $d$?
In general, you have $d(L_1, L_2) = \Vert P_{L_1} - P_{L_2} \Vert = \Vert \sin \Theta \Vert$, where $\Vert \cdot \Vert$ is the operator norm on matrices in $\mathbb{R}^{n \times n}$ and $\Theta$ is the diagonal matrix of the principal angles $\theta_1, \theta_2, \dots, \theta_k$ between the two subspaces (e.g. see Wikipedia also this post). One way to realize $\Theta$ is the following: let $A, B \in \mathbb{R}^{n \times k}$ be two matrices containing orthonormal bases for $L_1$ and $L_2$ respectively as their columns. If $\sigma_i$ are the singular values of $A^T B$, then $\sigma_i = \cos \theta_i$.
This characterisation gives a very geometric interpretation of the distance. For example, if $L_1$ and $L_2$ are lines, i.e. spanned by the unit vectors $v$ and $w$ say, then $\theta = \cos^{-1}(u^T w)$ and hence $d(L_1, L_2) = |sin(\theta)| = \sqrt{1 - \cos^2(\theta)} = \sqrt{1 - (u^T w)^2}$.