In a paper that I am reading, the author is weighting edges in a graph using

$w_k \propto det(D(p))$

where $D(p)$ is the metric tensor (which if I understand correctly is a space-varying metric?). They say that $D(p) = 1$ is the Euclidean metric, $D(p) = \mbox{const}$ is a Riemannian metric. Can anyone explain how to interpret what the determinant of a metric means? And can you suggest (in layman's terms) a metric which is not constant in position?

I see $det(D)$ so I think $D$ must be a matrix - is this correct? What is the matrix for a few simple metrics?

Thanks,

Dave