  We study diffusion processes in regions generated by sliding a cross section
by the phase flow of vector filed on curved spaces of arbitrary dimension. We
do this by studying the effective diffusion coefficient D that arises when
trying to reduce the n-dimensional diffusion equation to a 1-dimensional
diffusion equation by means of a projection method. We use the mathematical
language of exterior calculus to derive a coordinate free formula for this
coefficient in both infinite and finite transversal diffusion rate cases. The
use of these techniques leads to a formula for D which provides a deeper
understanding of effective diffusion than when using a coordinate dependent
approach.
