Local vs distance function metric structures

Question

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?

Answer 1

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 $\mathbb C$.)

Answer 2

I think the book on Petrunin's homepage might be of interest to you.

http://www.math.psu.edu/petrunin/papers/alexandrov/bbi.pdf

The part of the Finsler metric answer your question in the positive.