It is well-known that the length of a curve $\gamma: I=[a,b] \to S$, where $S \subseteq \mathbb{R}^3$ is a surface, may be computed by the first fundamental form as follows:
$$ L(\gamma)=\int_{a}^b\sqrt{E(u(t),v(t))\frac{du}{dt}^2+2F(u(t),v(t))\frac{du}{dt}\frac{dv}{dt}+G(u(t),v(t))\frac{dv}{dt}^2} dt $$
Conversely, suppose that we are able to compute lengths of curves, i.e. suppose that given $p \in S$, $v \in T_p(S)$ and $\gamma: (-\delta,\delta) \to S$ a curve such that $\gamma(0)=p$ and $\gamma'(0)=v$, we denote by $l(\gamma)$ the length of the curve $\gamma$. Is it possible to get from these data the first fundamental form $I_p(v)$?