While thinking about foliations I thought about a property that a pair of manifolds could have, which I will call "foliational reciprocity."
Question:
Does there exist a pair of smooth manifolds $(M,N),$ $M\ne N$ related by an isometry $g,$ s.t. every smooth regular foliation of $M$ satisfies some differential equation on $N,$ and every smooth regular foliation of $N$ satisfies some differential equation on $M?$
EDIT 7/5/2023:
This is an attempt to clarify the statement: "every smooth regular foliation of $M$ satisfies some differential equation on $N$."
Let me start by saying that there's a bijection between the set of foliations of $M$ and the set of differential equations that those foliations satisfy. This is due to the fact that foliations by definition satisfy differential equations. Given a foliation I can give you a differential equation whose solutions are the leaves of the foliation.
So the question asks whether there can exist a one-to-one correspondence between foliations on $N$ and the class of differential equations on $M$ which are satisfied by foliations of $M$ already. Moreover the reciprocity comes into play when I ask whether the reverse is simultaneously true i.e. whether there can exist a one-to-one correspondence between foliations on $M$ and the class of differential equations on $N$ which are satisfied by foliations of $N$ already.
Essentially one is taking all smooth regular foliations on $M,$ restricting to the metric of $N$ which can be obtained via the isometry $g,$ and checking whether these foliations satisfy a set of differential equations. The same process is done with foliations on $N.$
If we consider $(M,h)$ with metric $h$ and the set of foliations of $M$ then the set of foliations of $M$ satisfy a particular type of differential equations. This is the same case for $(N,j)$ with metric $j$ achieved through the isometry. And then one asks whether swapping the foliations but fixing the same metric results in each foliation satisfying a differential equation.
In this question I showed in one direction that a special foliation of $\Bbb R^2_+/\lbrace 1 \rbrace$ satisfies a heat equation on $\Bbb R^2$ with additional diffusivity parameters depending on space and time. As I wrote in that post, I achieved this by prescribing a foliation on $\Bbb R^2/\lbrace 0 \rbrace$ and using the $\exp$ isometry to transport the metric and that foliation onto the manifold $\Bbb R^2_+/\lbrace 1 \rbrace.$ Then I restricted the metric back to the usual Euclidean metric and checked that the foliation indeed satisfied the heat equation I mentioned.
I only showed one direction however, not both directions. Additionally this is only one foliation there are many more possible ones on $\Bbb R^2/\lbrace 0 \rbrace.$ Of course one might need to look outside flat Euclidean space to find a pair $(M,N).$