Fix two positive integers $k,n$ with $1 < k < n$ and equip the Grassmannian $\mathbf{Gr}(k,n)$ with your favorite metric $\rho$. Now fix $\delta > 0$ and consider an open ball $B_{\delta}(x)$ about some $x\in \mathbf{Gr}(k,n)$, defined in terms of this metric.

I'd like to know when $B_{\delta}(x)$ can be interpreted/realized as an open neighborhood in the space of lines $\mathbf{Gr}(1,n)$. Concretely, if I let $\rho_1$ denote some metric defined for $\mathbf{Gr}(1,n)$, I'd like to know when I can write something like $$B_{\delta}(x) "=" B_{\delta_1}(P),$$ where the right-hand side of this "equality" is to be interpreted as the Hausdorff distance to some suitable set $P \subset \mathbf{Gr}(1,n)$ and $\delta_1>0$ specifies the ball/distance in the $\rho_1$ metric. Generally speaking I would guess that there is a linear relationship between $\delta$ and $\delta_1$, with the constant of proportionality depending only on the choice of metrics $\rho,\,\rho_1$.

PS for my particular interests I think it would be good enough to have a characterization for the cases $\mathbf{Gr}(1,n)$ and $\mathbf{Gr}(2,n)$.



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