# Approximation of continuously varying Riemannian metric

As a part of my current studies, I have come up with the following problem. I believe that its solution may take quite some time. So before diving in I would like to know whether the similar problem is already known in literature so I don't have to reinvent anything. It may also be the case that the whole setup is meaningless for some(unknown to me) reason. I would appreciate if someone could give me any hints about this problem.

So here is the setup. I would like to approximate continuously varying Riemannian metric by number of patches with some constant Riemannian metric on each. As a particular example let us take metric of hyperbolic disc: $$ds^2 = l_{\text{AdS}}^2(d\rho^2 + \text{sinh}^2\rho d\theta^2) \quad (1)$$ here $l_{\text{AdS}}$ is constant which defines value of curvature. $\rho \in (0, +\infty)$ is radial coordinate and $\theta \in (0, 2 \pi)$ is angle coordinate.
I would like to project this metric on $R^2$. Then fill $R^2$ with rectangles of constant size $\delta^2$(one would eventually want to set $\delta \to 0$). In each region we define constant metric by averaging hyperbolic metric in it.

Now in order to see whether approximation works well, I would like to consider minimal geodesic between two arbitrary points. First, we obtain its length from (1) and then compare it with the one which comes from the approximation. Ideally one would like to show that for $\delta \to 0$ two expressions for the length are equivalent to any geodesic.

Another natural question to ask is whether this approach works for arbitrary continuously varying Riemannian metric? At this point, I can only see that this approach won't work at least if given Riemannian manifold can't be "properly" projected on top of $R^2$. For instance, this approach won't work for Klein bottle. Otherwise, I see no obstacles.

So if anyone has ever confronted with some articles or books which contain any information about the subject please let me know. If there is also some argument which may prove this approach to be wrong I would like to know about it. Once I come up with some conclusion I hope to post it here.

• What do you mean by a "constant" Riemannian metric? Do you mean "flat metric"? If so, then yes, there are standard approximation procedures when the approximating metric is piecewise-flat (it is singular at the vertices, of course). – Moishe Kohan Feb 27 '18 at 0:43