Is this a metric on $\mathbf P\mathcal H$?

Question

Let $\mathcal H$ be a real or complex Hilbert space with inner product $\langle\cdot,\cdot\rangle$. On the projective space $\mathbf P\mathcal H = \left(\mathcal H\setminus\{0\}\right)\big/{\sim}$ define \begin{align*} d\colon \mathbf P\mathcal H \times \mathbf P\mathcal H &\longrightarrow \mathbb R_{\ge 0},\\ ([v], [w]) &\longmapsto \cos^{-1} \left| \left\langle \frac{v}{\|v\|}, \frac{w}{\|w\|} \right\rangle \right| \quad\in\quad\left[0,\frac{\pi}{2}\right]. \end{align*}

Does $d$ satisfy the triangle inequality, i.e. is $d$ a metric on $\mathbf P\mathcal H$?

Does this construction have a name? Any references are welcome!

Answer 1

The metric you described is the standard metric on the projective space: in the real case it can be visualized as the angle between lines (thinking of the elements as lines). It arises as the quotient of the spherical metric on $S^n$ by the group of isometries $\{x\mapsto \alpha x, \ |\alpha|=1\}$ where $\alpha$ belongs to the ground field, $\mathbb{R}$ or $\mathbb{C}$. Indeed, $$ \left| \left< \frac{v}{\|v\|}, \frac{w}{\|w\|} \right> \right|=\sup_{|\alpha|=1} \operatorname{Re} \left< \frac{\alpha v}{\|v\|}, \frac{w}{\|w\|} \right>$$ and since $\cos^{-1}$ is decreasing, $$\cos^{-1} \left| \left\langle \frac{v}{\|v\|}, \frac{w}{\|w\|} \right\rangle \right| = \inf_\alpha \cos^{-1}\operatorname{Re} \left< \frac{\alpha v}{\|v\|}, \frac{w}{\|w\|} \right> = \inf_\alpha \rho \left( \frac{\alpha v}{\|v\|}, \frac{w}{\|w\|} \right)$$ where $\rho$ is the intrinsic (arclength) metric on $S^n$.

It's probably best to prove the triangle inequality as a special case of the general fact: whenever $G$ is a group acting on a metric space $(X,\rho)$ by isometries, the quotient $X/G$ has the quotient metric $$ d([x],[y]) = \min_{f\in G}\rho (f(x),y) $$ This is proved, e.g., in A course on metric geometry by Burago-Burago-Ivanov. Also discussed here.

This metric is also mentioned in metric and measure on the projective space. I would call it "the quotient metric" or "the canonical metric" on the projective space.