Discrete approximations of Riemannian geometry

Question

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.)

Answer 1

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

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

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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."