$\DeclareMathOperator\SL{SL}$Fix an integer $p\geq 1$ and a cocompact lattice $\Gamma\subset \SL(p+1,\mathbb{R})$. Consider the manifold
$$
M\_{\Gamma}:=\SL(p+1,\mathbb{R})/\Gamma.
$$
Let $A\subset \SL(p+1,\mathbb{R})$ be the subgroup of diagonal matrices with positive entries. The action
$$
A\curvearrowright M\_{\Gamma};\ a\cdot x\Gamma=(ax)\Gamma
$$
is called the **Weyl chamber flow**. Denote by $\mathcal{F}$ the orbit foliation on $M\_{\Gamma}$.
Is $\mathcal{F}$ Riemannian?



Is the orbit foliation of the Weyl chamber flow Riemannian?
These foliations are very far from being Riemannian.


Consider the case $p=1$, and take $PSL(2,\mathbb R)$ instead of $SL(2,\mathbb R)$ (just to simplify a bit). I'll explain how to construct an example where one non-compact leaf converges to a compact one (a circle). Such behavior implies that the foliation is not Riemannian.


Let $\Gamma\_g\subset PSL(2,\mathbb R)$ be the fundamental group of a genus $g\ge 2$ surface. You can think of $PSL(2,\mathbb R)$ the unit tangent bundle to $\mathbb H^2$ (the hyperbolic plane). In fact $SO(2)\setminus PSL(2,\mathbb R)$ can be identified with $\mathbb H^2$. Then $\mathbb H^2/\Gamma\_g$ is a compact hyperbolic surface $S\_g$. The action of the diagonal group on $PSL(2,\mathbb R)/\Gamma\_g$ is the geodesic flow on the unit tangent bundle of $S\_g$. Now, this flow has closed orbits - corresponding to closed geodesic on $S\_g$. Take such a closed geodesic $\gamma$, then it's not hard to construct a different geodesic that converges to $\gamma$. To see how this looks like, consider, for example Fig 3, in <https://arxiv.org/pdf/1610.07409.pdf> .


