a distance between $\mathbf{A} \in \operatorname{Gr}(k,n)$ and $\mathbf{B}\in \operatorname{Gr}(l,n)$ where $k \ne l$ is to first isometrically embed $\operatorname{Gr}(k,n)$ and $\operatorname{Gr}(l,n)$ into an ambient Riemannian manifold and then define the distance between $\mathbf{A}$ and $\mathbf{B}$ as their distance in the ambient space. This approach is taken in [@CHS; @SLLM], via an isometric embedding of $\operatorname{Gr}(0,n), \operatorname{Gr}(1,n), \dots, \operatorname{Gr}(n,n )$ into a sphere of dimension $(n-1)(n+2)/2$. Such a distance suffers from two shortcomings: It is not intrinsic to the Grassmannian and it depends on both the embedding and the ambient space.

The distance that we propose in Section \[sec:main\] will depend only on the intrinsic distance of the Grassmannian and is independent of $n$, i.e., a $k$-plane $\mathbf{A}$ and an $l$-plane $\mathbf{B}$ in $\mathbb{R}^n$ will have the same distance if we regard them as subspaces in $\mathbb{R}^m$ for any $m \ge \min(k,l)$. We will first establish this for the special case $k = l$.

Consider the inclusion map $\iota_n:\mathbb{R}^n\to \mathbb{R}^{n+1}$, $\iota_n(x_1,\dots, x_n)=(x_1,\dots, x_n,0)$. It is easy to see that $\iota_n$ induces a natural inclusion of $\operatorname{Gr}(k,n)$ into $\operatorname{Gr}(k,n+1)$ which we will also denote by $\iota_n$. For any $m > n$, composition of successive natural inclusions gives the inclusion $\iota_{nm} : \operatorname{Gr}(k,n) 