Asymmetric Geometry of Total Grassmannians

Abstract

Metrics in Grassmannians, or distances between subspaces of same dimension, have many uses, and extending them to the Total Grassmannian of subspaces of different dimensions is an important problem, as usual extensions lack good properties or give little information. Dimensional asymmetries call for the use of asymmetric metrics, and we present a natural method to obtain them, extending all the main Grassmannian metrics (geodesic, projection Frobenius, Fubini-Study, gap, etc.). Their geometry adequately reflects containment relations of subspaces, continuous paths link subspaces of distinct dimensions, and we describe minimal geodesics, shortest paths to move a subspace onto another. In particular, the Fubini-Study metric extends as an asymmetric angle that is easily computed, has many useful properties, and a nice geometric interpretation.

Figures (11)

• Figure 1: Backward balls of small radius $r$ around a line $L$ and a plane $V$ in $\mathop{\mathrm{Gr}}\nolimits^-(\mathds{R}^3)$
• Figure 2: Balls $B_r^-(V)$ stretch downwards to include neighborhoods of $V'\subset V$, and likewise for W, so $\gamma_1$, $\gamma_2$ and $\gamma_3$ form a continuous curve. In $\mathop{\mathrm{Gr}}\nolimits^+(n)$, balls stretch upwards, so closed dots would be below. Moving downwards causes a length change $\Delta L = d(W,W') = \Delta_q$, while upwards $\Delta L = d(V',V)=0$ (see \ref{['pr:length partition']}).
• Figure 3: Topologies of the Total Grassmannian. The map $V\mapsto V^\perp$ is a homeomorphism of $\mathop{\mathrm{Gr}}\nolimits(n)$, and flips the asymmetric $\mathop{\mathrm{Gr}}\nolimits^{-}(n) \leftrightarrow \mathop{\mathrm{Gr}}\nolimits^{+}(n)$, so the pair remains symmetric. In $\mathop{\mathrm{Gr}}\nolimits^\pm(n)$, the $\mathop{\mathrm{Gr}}\nolimits_p(n)$'s are much more interconnected than shown.
• Figure 4: Type I and II paths between a line $V$ and a plane $W$ in $\mathds{R}^3$.
• Figure 5: Subspaces and paths used in \ref{['pr:I II exist']}(\ref{['it:L1']}). Dashed lines mean $\DOTSB\mathbin{\vcenter{\hbox{$\m@th\boxplus$}}}$. The type I path $\gamma = \mu \DOTSB\mathbin{\vcenter{\hbox{$\m@th\boxplus$}}} \eta : V_{(p)} \leadsto W$ combines $\mu: V \overset{p}{\hookrightarrow} W'$ formed by $\mu_1 : V \overset{p}{\hookrightarrow} U'$ and $\mu_2:U' \overset{p}{\hookrightarrow} W'$, and $\eta:0 \dashrightarrow T$ formed by $\eta_1: 0 \dashrightarrow R$ and $\eta_2:R \nearrow T$.

Theorems & Definitions (137)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Definition 2.9
• Lemma 2.10