Does a $1$-holomorphic form on a Riemann surface induce a metric on that surface?

Question

Does a $1$-holomorphic form $\omega = f(z)\, dz$ on a Riemann surface $S$ induce a sense of distance on the surface? As, induces a Riemannian metric?

Answer 1

Yes, as long as $\omega$ is nowhere zero. The real $2$-tensor field $g = \omega\cdot\overline\omega$ is a Riemannian metric (where the dot represents the symmetric product, $\omega\cdot\overline\omega = \tfrac12 (\omega\otimes\overline\omega + \overline\omega \otimes\omega)$). In any holomorphic coordinate chart $z=x+iy$, it is $g= |f|^2(dx^2 + dy^2)$.