  Several aspects of managing a sensor network (e.g., motion planning for data
mules, serial data fusion and inference) benefit once the network is linearized
to a path. The linearization is often achieved by constructing a space filling
curve in the domain. However, existing methods cannot handle networks
distributed on surfaces of complex topology. This paper presents a novel method
for generating space filling curves for 3D sensor networks that are distributed
densely on some two-dimensional geometric surface. Our algorithm is completely
distributed and constructs a path which gets uniformly, progressively denser as
it becomes longer. We analyze the algorithm mathematically and prove that the
curve we obtain is dense. Our method is based on the Hodge decomposition
theorem and uses holomorphic differentials on Riemann surfaces. The underlying
high genus surface is conformally mapped to a union of flat tori and then a
proportionally-dense space filling curve on this union is constructed. The
pullback of this curve to the original network gives us the desired curve.
