Two metrics on $\mathbb P^n(\mathbb C)$

Question

A far as I know there are two different ways to construct a "distance function" on the complex projective space $\mathbb P^n(\mathbb C)$. I would like to understand their relationship:

1. Chordal distance: For $x=(x_0:\ldots:x_n)$ and $y=(y_0:\ldots: y_n)$ one can define: $$d_{ch}(x,y):=\frac{\left(\sum_{i<j}|x_iy_j-x_jy_i|^2\right)^{1/2}}{\|x\|\,\|y\|}$$ where $\|\cdot\|$ denotes the standard euclidean norm in $\mathbb C^{n+1}$. With a bit of work one can show that $d_{ch}(x,y)$ is a well defined distance. It is called "chordal" because if we consider $P^1(\mathbb C)$ embedded as the Riemann sphere inside $\mathbb R^3$, then $d_{ch}(x,y)$ should be exactly the length of the chord joining $x$ and $y$.

2. The Fubini-Study metric: As smooth real manifolds, we have the identification $\mathbb P^n(\mathbb C)\cong S^{2n+1}/S^1$. On the unit hypershere $S^{2n+1}$ we can put the standard round Riemannian metric (aka the pullback of the euclidean metric) and then we can push this metric on the quotient $S^{2n+1}/S^1$ i.e. we let $S^1$ (=group of rotation) act diagonally on $S^{2n+1}$. Actually, the best reference I know about the construction of the Fubini-Study (Riemannian) metric is the wikipedia page. There you can really find everything, even the explicit expression in terms of local coordinated with respect to the standard affine patches of $\mathbb P^n(\mathbb C)$. So, lets denote with $g$ the tensor field of the Fubini-Study metric, then $(\mathbb P^n(\mathbb C),g)$ is a beautiful Riemannian manifold (or hermitian manifold if considered over $\mathbb C$). At this point $g$ induces on $\mathbb P^n(\mathbb C)$ a distance function $d_{fs}(x,y)$.

My question is the following:

What is the relationship between the functions $d_{ch}$ and $d_{fs}$? My intuition is that they should be infinitesimally the same (even if I cannot prove it formally). If we oversimplify the picture and we imagine we are working on a circle, then $d_{ch}$ measures the chord and $d_{fs}$ measures the arch lying on that chord. Unfortunately, very often I've seen people saying sentences like: "the cordal distance on $\mathbb P^n(\mathbb C)$ induces the Fubini-Study metric". So, probably I am missing something and these two metric are really closely related to each other; more than I think!

Answer 1

The chordal metric is a "topological metric" in the sense of metric spaces, while the Fubini-Study metric is a "Riemannian metric" in the sense of differential geometry. To compare apples to apples, we can look at the topological "arc length" metric induced by Fubini-Study, defined by lengths of shortest geodesics.

On the sphere, the chordal metric is smooth at the antipode, while Fubini-Study is not. The two metrics are asymptotically close in a neighbrhood of each point, however. The diagram shows the graphs of the respective distance functions over an equatorial circle. (Politeness might have dictated squashing the vertical axis rather than plotting to scale, but the true shape of the two helices was too visually pleasing.)