Let $M$ be a smooth $n$-dimensional manifold. I have heard that a Riemannian metric on $M$, depends locally on $ n(n+1) / 2 - n = n(n-1) /2$ "independent" functions up to coordinate changes.
I can roughly see the intuition for that: The symmetric matrix $g_{ij} $ depends on $n(n+1) / 2$ functions, but you can "change the coordinates" which loses $n$ degrees of freedom.
Is there a precise formulation of this statement? Given an arbitrary metric, can you really specify $n$ out of the $ n(n+1)/2$ $g_{ij}$ functions as you wish? How can we see it's impossible specify more than $n$?
I heard that Cartan proved something like this, using the language of jets. (Something about the dimension of the space of $k$-jets of metrics modulu diffeomorphisms).
Maybe there are other references; any pointer in the right direction is welcomed.
This question was supposedly answered here, but I find the answer there to be too vague. I am looking for a precise statement.