Wikipedia:Reference desk/Archives/Mathematics/2021 January 30

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January 30

Metric-space manifold where you "can keep going"

Is there a standard name for a metric space for which there is some fixed ε and some dimension n such that, around every point, the n ball of radius ε is "approximately isometric" (I haven't worked out what that means exactly) to the Euclidean n-ball of radius ε?

The question arose in the context of trying to figure out what it means to say that the physical universe might be "finite but unbounded". If it's finite, then it's obviously not "unbounded" in the metric-space sense. Saying it's "without boundary" (as opposed to a manifold with boundary) gets closer, but doesn't quite capture it; for example, an open 3-ball is also a manifold without boundary, but in a metric-space sense, it comes to an end. The notion seems to be specifically metric, not just topological, but "unbounded" is clearly the wrong word from a mathematician's perspective. --Trovatore (talk) 05:50, 30 January 2021 (UTC)

A punctured sphere is topologically equivalent to a plane, so clearly the metric is essential; otherwise adding a point to an infinite space could make it finite. If we restrict ourselves to finite-dimensional metric spaces, then, I think, the mathematical equivalent of the intuitive everyday sense of "finite" is that it can be covered by a finite number of n-balls of some given radius, which is equivalent to being bounded in the topological sense. So what we need is a mathematical translation of the intuitive notion of "unboundedness", which I think means that a space traveller will never run into some intransgressable boundary. That would correspond (I think) to a kind of Archimedean property – a traveller can keep travelling over arbitrary long distances; although they may return to their point of departure, there were no obvious shortcuts on the way like you have with the Peano curve. Would this work?: From any given point of departure x and a point y at a distance ε, consider how far from x one can get by travelling another distance ε from y, arriving at z. This should then be almost 2ε apart from x, so we may require that the limit of d(x, z)/ε, as ε tends to 0, equals 2.  --Lambiam 10:23, 30 January 2021 (UTC)

Oh, I was very close to a precise definition already, just needed to fill in what "approximately isometric" means. I was thinking something like, there's a bijection f between the two n-balls such that the distance between f(x) and f(y) is between (1-ε)d(x, y) and (1+ε)d(x, y). The details probably don't matter much because you can always pick a smaller epsilon. The point is that the relative error should be order ε.

Anyway, the question wasn't so much about the formulation as it was about whether there's already a name for this or something very similar to this. --Trovatore (talk) 18:57, 30 January 2021 (UTC)

Mmm, probably want one more criterion, which is that the approximate isometries cohere in the overlap of nearby &epsilon-balls. --Trovatore (talk) 19:03, 30 January 2021 (UTC)

From a quick read, not sure exactly what you are asking for. But diameter for a metric space, Riemannian manifold - especially the sections on diameter, Hopf-Rinow, (geodesic) completeness might help.John Z (talk) 19:11, 30 January 2021 (UTC)

Ah, maybe I just want to say, manifold without boundary that's complete as a metric space. That does exclude the open ball (but allows R³) and is a lot more concise. Thanks, John Z. --Trovatore (talk) 19:37, 30 January 2021 (UTC)

Actually, now there's an interesting math question, as opposed to just terminology. Suppose you have a Riemannian 3-manifold (without boundary) that's complete as a metric space. Does it follow that it has the property I elucidated? That there's some fixed ε such that every point has an ε-ball around it that's approximately isometric to the Euclidean 3-ball, and these neighborhoods cohere in some appropriate sense, analogous to charts and atlases in the definition of smooth manifold? --Trovatore (talk) 21:13, 30 January 2021 (UTC)

Mmm, probably not, because you could take a homeomorph of R³ and put a metric on it where there are lots of little regions of high curvature, going arbitrarily high. Maybe if we assume the manifold is compact? --Trovatore (talk) 21:17, 30 January 2021 (UTC)

Consider expansion from a given point of departure by the repeated Minkowski addition of ε-balls. What will it do on this space? Physical beings are constrained from exploring regions whose curvature exceeds a certain limit, so SF writers should be content with definitions assuming bounded curvature, but the mind can go where no creature can follow it.  --Lambiam 17:41, 31 January 2021 (UTC)

Update: Following a thread from John Z's links, I came across geodesic manifold, which looks like exactly what I was looking for. The Hopf–Rinow theorem doesn't exactly answer the question I asked on 21:13 30 January, but it's in the ballpark. --Trovatore (talk) 19:12, 1 February 2021 (UTC)