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Let $n=3.$

Take $X=(0,1)^n.$ Fix points $p,q$ s.t. $\text{dist}_n(p,q)=\sqrt{n}.$ Construct a smooth regular foliation of $X$ with $(n-1)-$dim. leaves which are topologically $(0,\sqrt{n})\times S^{n-2} $ accumulating to $p,q.$

In this 2021 paper by Inaba and Masuda Foliations on the open 3-ball by complete surfaces they describe classes of surfaces that foliate $X=(0,1)^3$ without any singularities. One class they describe is $S=\Bbb R^2 - (C \times \lbrace{0}\rbrace)$ where $C$ is the Cantor set. Another class is $S=\Bbb R^2-(\Bbb Z \times \{0\}).$

My formulation reduces to the class $S=\Bbb R^2- (\text{1 point})$ which the authors do not consider. It's curious that Inaba and Masuda have found weirder examples than mine which do foliate $X=(0,1)^3$ without any singularities. This suggests that it might be possible to achieve such a foliation.

Note that the leaves in $X$ are all diffeomorphic to open annuli, and the open annuli are diffeomorphic to once punctured planes. This is why the problem is equivalent to asking about a foliation by once punctured planes.

How do you construct a foliation of $X$ with the class $S=\Bbb R^2- (\text{1 point})$ with no singularities?

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  • $\begingroup$ The supremum over all distances between two points of (0, 1)^n is √n, but this distance is not achieved by any p, q ∊ (0, 1)^n. So I have to wonder what set your p and q are members of. $\endgroup$
    – Daniel Asimov
    Commented Mar 29, 2023 at 17:37
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    $\begingroup$ @DanielAsimov The leaves in $X$ are all diffeomorphic to open annuli, and the open annuli are diffeomorphic to once punctured planes. So the question is really how to foliate $X$ with once punctured planes (without any singularities present). As for your question about $p,q$ - for $n=3,$ an example of the points $p,q$ is: $p=(0,1,1)$ and $q=(1,0,0).$ Another example is $p=(0,0,0)$ and $q=(1,1,1).$ $\endgroup$
    – John McManus
    Commented Mar 29, 2023 at 19:57
  • $\begingroup$ Normally (a, b) is the open interval, so (0, 1)^3 is an open unit 3-cube. My question was: What set or sets are p and q members of? (Not: "What are examples of p and q?") $\endgroup$
    – Daniel Asimov
    Commented Mar 29, 2023 at 23:26
  • $\begingroup$ @DanielAsimov In general dimension $n$ you could say the points $p,q$ are particular pairs of the set of all vertices on the $n$-cube. The prescription I gave for those pairs is that $\text{dist}_n(p,q)=\sqrt{n}.$ If I relax this and require only that $p,q$ are some pair of distinct vertices on the $n$-cube (all else equal), then the problem still is equivalent to a foliation of $X$ by once punctured planes (in dimension $n=3$) and corresponding higher dimensional equivalents. So the formulation I gave is simply another way of stating the problem. $\endgroup$
    – John McManus
    Commented Mar 31, 2023 at 16:47
  • $\begingroup$ Usually (0, 1)^n denotes the open n-cube and [0, 1]^n denotes the closed n-cube. The open version (0, 1)^n does not contain any of its boundary points. $\endgroup$
    – Daniel Asimov
    Commented Mar 31, 2023 at 17:45

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