# Realizing a neighborhood in $\mathbf{Gr}(k,n)$ as a neighborhood in $\mathbf{Gr}(1,n)$

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 $$\mathbf{Gr}(2,n)$$.