is identified with an equivalence class comprising all its $k$-frames $\{ AQ \in \operatorname{V}(k,n): Q \in \operatorname{O}(k)\}$. Note that $\operatorname{span}(AQ) = \operatorname{span}(A)$ for all $Q \in \operatorname{O}(k)$.

There is a *left action* of $\operatorname{O}(n)$ on $\operatorname{Gr}(k,n)$: For any $Q \in \operatorname{O}(n)$ and $\mathbf{A} = \operatorname{span}(A) \in \operatorname{Gr}(k,n)$ where $A$ is a $k$-frame of $\mathbf{A}$, the action yields $$\label{eq:action}
Q \cdot \mathbf{A}\coloneqq \operatorname{span}(QA) \in \operatorname{Gr}(k,n).$$ This action is transitive as any $k$-plane can be rotated onto any other $k$-plane by some $Q \in \operatorname{O}(n)$. A $k$-plane $\mathbf{A} \in \operatorname{Gr}(k,n)$ will be denoted in boldface; the corresponding italized letter $A =[a_1,\dots,a_k] \in \operatorname{V}(k,n)$ will denote a $k$-frame of $\mathbf{A}$.

$\operatorname{Gr}(k,n)$ and $\operatorname{V}(k,n)$ are smooth manifolds of dimensions $k(n-k)$ and $nk-k(k+1)/2$ respectively. As a set of $n\times k$ matrices, $\operatorname{V}(k,n)$ is a submanifold of $\mathbb{R}^{n\times k}$ and inherits a Riemannian metric from the Euclidean metric on $\mathbb{R}^{n\times k}$, i.e., given $A=[a_1,\dots, a_k]$ and $B=[b_1,\dots, b_k]$ in $T_X \operatorname{V}(k,n)$, the tangent space at $X \in \operatorname{V}(k,n)$, the Riemannian metric $g$ is defined by $g_X(A,B)=\sum_{i=1}^k a_i^\mathsf{T} b_i= \operatorname{tr}(A^\mathsf{T}B)$. As $g$ is invariant under the action of $\operatorname{O}(k)$, it descends to a Riemannian metric on $\operatorname{Gr}(k,n)$ and in turn induces a geodesic distance on $\operatorname{Gr}(k,n)$ which 