# Foliation of $X$ by once punctured planes without any singularities

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?

• 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. Commented Mar 29, 2023 at 17:37
• @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 $q=(1,1,1).$ Commented Mar 29, 2023 at 19:57
• 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?") Commented Mar 29, 2023 at 23:26
• @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. Commented Mar 31, 2023 at 16:47
• 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. Commented Mar 31, 2023 at 17:45