# Discrete approximations of Riemannian geometry

I am looking for the appropriate way to talk about a discrete geometry approaching a continuous Riemannian geometry.

By a discrete geometry I mean something like a graph or cell-complex, perhaps with additional data.

By approaching a continuous geometry I approximately mean something like the following. Take the discrete torus $(\mathbb{Z}/N)^2$, the $N\times N$ grid with opposite ends identified. In the limit $N\rightarrow\infty$ this clearly "approaches" in some sense the usual torus $T^2$, with the flat metric.

One way this approach happens is via the graph Laplacian on $\mathbb{Z}^2$. The spectrum of its lowest eigenvalues converges to the spectrum of the continuous Laplacian on $T^2$ as $N\rightarrow \infty$. Which is just a fancy way of saying that "coarse-grained" the two look alike.

This example of $\mathbb{Z}/N \rightarrow T^2$ has many non-generic properties that I do not wish to rely on, for example they both have a global translation symmetry and there is a natural embedding of the discrete geometry into the continuous.

To be more general, take a torus $M$ with a generic metric with bounded curvature - we can imagine the following discrete approximations to it:

1. The graph given by a locally optimal circle packing in $M$ with circle of radius $r$. As $r\rightarrow 0$ this should "converge" to $M$.

2. Given $d$ and $\epsilon$, Select $d^2$ random points according to the uniform distribution on $M$. Write the graph where two points $x$ and $y$ are connected if the geodesic distance between them on $M$ is less than $\epsilon$.

3. Select $d^2$ random points, and form the Voronoi diagram on $M$, and label each cell by it's area. This gives a cell complex which "approximates" $M$.

Has anyone studied, and under what name, these sorts of convergence problems? Is there more to be said than graph spectral theory.

What I would be most interested in is if for some discrete data like this, there were a direct discrete analog of the usual objects of Riemannian geometry. For example, is there some way to attach to a graph a "discrete Christofel symbol" to a graph? Just to throw something out there: parallel transport maps the tangent space of point $a$ to the tangent space of point $b$ given a curve from $a$ to $b$, so perhaps the "discrete Christofel symbol", maps a linear combination of edges connected to $a$ to a linear combination of edges connected to $b$.

(Incidentally there is a question titled Is there such a thing as discrete Riemannian geometry? but it seems to be unrelated.)

• yann-ollivier.org/rech/publs/visualcurvature.pdf The second part of this may be related. Unfortunately, I have not yet studied it so I cannot know to what extent. I'm sorry if it's a waste of time. Aug 11, 2017 at 22:27
• A lot of research in DDG has been done by the computer graphics community, as a result there are often many results that don't have rigorous convergence arguments (which is something that it seems like you are interested in). You might be interested in the recent paper by Schumacher & Wardetzky titled Variational Convergence of Discrete Minimal Surfaces. I mention this paper since it shows the power of the discrete framework, and since I think the variational convergence framework introduced will be very powerful tool for showing the convergence of many other geometric functionals. Aug 15, 2017 at 7:52

This is an active research area recently, e.g.,

Bobenko, Alexander I., ed. Advances in Discrete Differential Geometry. Springer, 2016. (Springer link).

Here's a short essay (a special issue introduction) on the topic, with 36 references:

Gu, David, and Emil Saucan. "Discrete Geometry—From Theory to Applications: A Case Study." (2016): MPDI Axioms 27. (MPDI link.)

They mention Ollivier's approach to Ricci curvature in a graph, as cited also by Alessio Di Lorenzo in a comment, although ultimately they lobby for "Forman curvature."