I have a piecewise linear (PL) surface transversely immersed in $\mathbb{R}^3$; is this a Riemann surface in the sense that I can describe it with a local coordinate $z\in \mathbb{C}$? My basic argument is that in 2 dimensions I think the PL and smooth categories coincide, so the question reduces to "can any smooth immersed surface in $\mathbb{R}^3$ be a Riemann surface?" My first instinct is that this is false; the added complex structure would chance the nature of the surface (ie the Cauchy-Reimann equations must now be satisfied). However, you can put an almost complex structure on any even-dimensional real manifold so I'm thinking the statement might actually be true.

Any ideas?