Consider a real analytic manifold $(M,g_1,\mathfrak{A})$ with metric $g_1$ and real analytic foliation, $\mathfrak{A}.$ And consider $(N,g_2,\mathfrak{B})$ with analytic diffeomorphism $f:M \to N,$ for $\dim(M)=\dim(N).$ Take the foliation $\mathfrak{B}$ and $g_1$ and form the object $(O,g_1,\mathfrak{B}),$ where $O$ is a properly embedded submanifold in $M.$ Due to the embedding, $O$ inherits the induced metric $g_1.$
An example I think could work is: $M:=\Bbb R^2\setminus\{0\}$ with $g_1$ the usual Euclidean metric, and $\mathfrak{A}$ the foliation by rectangular hyperbolae, with $f$ the $\exp$ mapping. You can push the metric from $M$ onto $N:=\Bbb R^2_+/\lbrace1\rbrace$ where you have $ds^2=dx^2+dy^2$ becoming $ds^2=\frac{du^2}{u^2}+\frac{dv^2}{v^2}.$ Then you can take the foliation on $N$ and embed it properly back into $M$ in a way that you preserve the foliation from $N$ and you preserve the metric from $M$ and package them both into one manifold.
The significance of this process is that it yields a solution to the heat equation (for the example I sketched) with $\mathfrak B$ serving as the particular solution. This is surprising because it's not immediately apparent why the foliation on $N$ should solve the heat equation when $g_2 \mapsto g_1$ is applied. Note that $\mathfrak{A}$ does not solve the heat equation.
In the commments it is argued that $O$ and $\mathfrak B$ are not related. Then my question becomes:
If $O$ and $\mathfrak B$ are unrelated, then why does $\mathfrak B$ solve an important partial differential equation written in terms of the metric $g_1?$
Doesn't this imply that they are related since $\mathfrak B$ carries information about the solution to a differential equation on $O?$
Let's be a little more precise. We have the heat equation (with some additional diffusivity parameters):
$$ x u_{xx}= \mp t u_t $$
where our foliational solution is:
$$\mathfrak B=\big \lbrace e^{\pm \frac{x}{\log t}}: x>0, t\ne 1\big \rbrace $$
Is there a concrete reason why this process (for my sketched example) works to solve this heat equation?