Kodaira's Complex Analysis defines a metric for Riemann Surface by using partition of unity:
Let $\mathcal{R}$ be an arbitrary Riemann surface. Let $\mathfrak{W} = \{W_n\}$ be a locally finite open covering consisting of coordinate disks $W_n$, let $w_m$ be a local coordinate on $W_m$ and let $\{\rho_m\}$ be a partition of unity subordinated to $\mathfrak{W}$. For $p, q \in \mathcal{R}$ we define the distance $d(p,q)$ between $p$ and $q$ by
$$ d(p,q) = \sum_m \big(|\rho_m(p) - \rho_m(q)| + |\rho_m(p) w_m(p) - \rho_m(q) w_m(q)|\big) $$ Here we assume the $\rho_m (p)w_m(p)$ has been extended to a $C^\infty$ function on $\mathcal{R}$ which vanishes whenever $p \notin W_m$.
Why here it sums two parts? What's the geometric intuition for this metric?
I think it should relate to the way constructing Riemannian metric on smooth manifold in the way mentioned here How to obtain a locally finite countable cover on a smooth manifold?. But I cannot get the metric 2-form from this distance function. Is these two related? I tried to define $\mathrm{d}s_m^2 = \rho_m \mathrm{d}x^2+\rho_m \mathrm{d} y^2$ but it does not give the above distance.
Any help would be appreciated. Thanks!
