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?



Determinant of a metric?
This is too long for a comment but still too short for an introductory course that you are asking for.


A Riemannian metric in the plane is (represented by) a matrix-valued function such that the value at every point $p$ is a symmetric matrix $g=g(p)$ of the form $g=\begin{pmatrix} E & F \\ F & G\end{pmatrix}$ which is positive definite as a quadratic form, that is, $E,G>0$ and $\det g >0$. Any matrix-valued function satisfying these properties is (represents) a Riemannian metric, by definition.


There are many examples. Usually introductions begin with metrics defined by parametric surfaces (that $r\_u$ and $r\_v$ stuff). This is only an example, one of the many, and it is irrelevant in the context of the paper. The paper considers Riemannian metrics not associated to any surface.


Let me try to explain what it is about. The above matrix $g$ defines a norm on $\mathbb R^2$ as follows: $\|(u,v)\|=\sqrt{Eu^2+2Fuv+Gv^2}$. This norm defines a metric $d$ by $d(x,y)=\|x-y\|$. This is a Riemannian metric associated with a constant metric tensor $g$. One can prove that this metric space is isometric to the standard Euclidean plane, and an isometry is given by a linear map (which can be written explicitly in terms of $E$, $F$ and $G$). So a constant metric tensor essentially produces a Euclidean metric written in some non-orthogonal coordinates.


What I said above is a reformulation of some bits from linear algebra. Riemannian geometry begins when $g$ is not constant. Then there is not a norm but a family of norms $\|\cdot\|\_p$, $p\in U$, where $U\subset\mathbb R^2$ is the domain where $g$ is defined. Every norm $\|\cdot\|\_p$ yields a metric $d\_p$ on its own private copy of $\mathbb R^2$ (this private copy is called the tangent space at $p$ and is usually denoted by $T\_pU$).


This family of metrics is "glued together" into one metric $d$ on $U$. It can be defined as the maximal metric satisfying the following condition: for every $p\in U$, one has
$$
\frac{d(x,y)}{d\_p(x,y)} \to 1 \ \ \ \text{as} \ \ x,y\to p .
$$
So, locally near a point $p$ the metric is approximately the same as the (essentially Euclidean) metric $d\_p$ defined by the matrix at $p$. This is sufficient for any first-order analysis.


The standard definition involves defining a length of a curve with respect to $g$ (integrate the length of the velocity vector, computed in that variable norm), and then defining the distance between points as the infimum of length of connecting curves. It is equivalent to the one I gave above.


