A manifold for Hilbert's hotel

Well, after recently answering a Hilbert's Hotel question, I've started to think: If an infinite number of people arrive, the solution is that every guest goes to the room with twice the number. However, if one imagines the doors one after the other (that is, in the order of the natural numbers), this means that each guest has to walk a distance that is proportional to the number of his room, which is unbounded. Assuming a constant walking speed, therefore also the move time is unbounded. That is not really a satisfying solution.

But this is easily solved: Just build the hotel in an infinite-dimensional Euclidean space, where room $n$ sits at the point which has coordinates $x_k=\delta_{nk}$ where $\delta$ is the Kronecker delta. That way, the distance between any two rooms is $\sqrt{2}$, and any room changing operation, no matter how complicated, can be done in constant time.

So far, so good. However, let's assume the guests in Hilbert's Hotel are conventional 3-dimensional beings, and therefore they must live on a 3-dimensional manifold; they would die in an unconstrained higher-dimensional space, let alone an infinite-dimensional one.

Therefore my question:

Does there exist a 3-dimensional Riemannian manifold which has a countably infinite number of points such that any two of them have the same finite distance on the manifold?

Solution 1:

If $x_n$ are such points then $R=d(x_1,x_n) >0$ for all $n$ Hence $x_n\in B_R(x_1)$ Since $B_R(x_1)$ is compact so there is a convergent subsequence. Hence it does not happen.

Solution 2:

For posterity here's an implementation of Micah's suggestion.

Let $(\rho, \theta, \phi)$ denote spherical coordinates on the open unit ball, $\Omega = d\phi^{2} + \sin^{2}\phi\, d\theta^{2}$ the round metric on the unit sphere, and $f(\rho) = \rho/(1 - \rho)$ (or any smooth, monotone function defined for $0 \leq \rho < 1$ with $f(0) = 0$ and, as $\rho \to 1^{-}$, $f \to \infty$ rapidly enough that $f$ is not improperly integrable).

In the metric $$ g = d\rho^{2} + f(\rho)^{2}\, \Omega, $$ the distance from the origin to the boundary of the ball is unity, but the volume element is $$ dV = f(\rho) \sin\phi\, d\rho\, d\theta\, d\phi. $$ Geometrically, the intrinsic radii of spherical shells centered at the origin grow rapidly enough that the volume is infinite.

In this universe, there exist countably many "cells" of fixed volume (though not suitable as three-dimensional hotel rooms, as asymptotically they necessarily become "intrinsically thin in the radial direction"), but any two rooms (or points) are separated by a distance of at most $2$ because the origin is at most one unit away from each room.