The geodesic distance $d(p\_1, p\_2)$ on a geodesically complete, Riemanian manifold is a metric in the sense of a metric space metric. I'd like to know when other infinitesimal metric structures (e.g. finlser metrics) induce such a global metric and what the exact relation of the two things is.


What is the largest class of Lagrangians $\mathcal{L}: TM \rightarrow \mathbb{R}$ such that the action it assigns to its stationary curves defines a distance function in the sense of metric spaces.


I'm also interested in any procedure that recovers a local metric structure (i.e. Riemanian tensor/Finsler function) from a given global metric $d:M \times M \rightarrow \mathbb{R}$. Any references about this would be helpful.


A few sources state that the Fubini-Study metric on $\mathbb{C}P^N$ can be "easily" recovered from the distance function $d(|\psi \rangle, |\phi\rangle) = \arccos(| \langle \psi |\phi \rangle|)$, what is the procedure for this and are these ideas rigorous?



Local vs distance function metric structures
I answer only your last question.


Take a finite-dimensional $\mathbb C$-linear space $V$ equipped with a positive-definite hermitian form $\langle-,-\rangle$. The tangent space $\text{T}\_p{\mathbb P}\_{\mathbb C}V$ at a $1$-dimensional linear subspace $p\subset V$ is naturally identified with $\text{Lin}\_{\mathbb C}(p,V/p)$ which in turn is naturally identified with $\text{Lin}\_{\mathbb C}(p,p^\perp)$ with respect to the orthogonal decomposition $V=p\oplus p^\perp$. Using this decomposition, we can assume that
$\text{Lin}\_{\mathbb C}(p,p^\perp)\subset\text{Lin}\_{\mathbb C}(V,V)$ (extending by zeros). The hermitian form on $\text{Lin}\_{\mathbb C}(V,V)$ given by the rule $\langle t\_1,t\_2\rangle=\text{tr}(t\_1\circ t\_2^\*)$ induces on
${\mathbb P}\_{\mathbb C}V$ the famous Fubini-Study hermitian structure. Naturally, the distance can be expressed in terms of the original hermitian form on $V$. The distance in the projective space between the points
$0\ne p\_1,p\_2\in V$ is given by
$\text{dist}(p\_1,p\_2)=\text{arccos}\frac{\langle p\_1,p\_2\rangle\langle p\_2,p\_1\rangle}{\langle p\_1,p\_1\rangle\langle p\_2,p\_2\rangle}$.


A similar approach works for many hyperbolic geometries (including the complex hyperbolic one), elliptic geometries, and many others that come from grassmannians.


**Late extension.** It is not necessary to require the form
$\langle-,-\rangle$ to be positive-definite, and one can also take $p\subset V$ of a fixed dimension, say, $k$, thus dealing with the grassmannian $\text{Gr}\_{\mathbb C}(k,V)$. As above, the loci of non-degenerate points get their pseudo-hermitian structure. Many objects of these geometries (such as geodesics, bisectors, etc) admit an easy linear/hermitian description. The degenerate points form the absolute whose geometry can also be described in the hermitian/linear terms. (It is also possible to take $\mathbb R$ or $\mathbb H$ (or even simetimes $\mathbb O$) in place of $\mathbb C$.)


