Explanation for the relation of a metric tensor in a vector space and the metric function on topological spaces

Question

I have always wondered, an it seems as I cannot figure it out on my own, the following.

Consider a differentiable manifold $M$ with a metric tensor $g$, which acts on tangent vectors at any point $p\in M $, now, obviously $g$ gives meaning to distances and angles in $M$, nevertheless $M$ is also a topological manifold, which is implies is second countable and Hausdorff, suppose it is also paracompact, then $M$ is indeed metrizable, which means there exists a map $d:M \times M \rightarrow \mathbb{R}$ that satisfies the following for any $x,y,z\in M$.

1. $d(x,y)\geq 0$, where the equality is satisfied if and only if $x=y$
2. $d(x,y)=d(x,y)$
3. $d(x,y)+d(y,z)\geq d(x,z)$

This map is called the distance function or metric (which makes me uncomfortable). My question is:

Is there a natural way to induce $d$ from $g$ or the converse?