On the Lie Foliation structure of Walker Manifolds

We study Walker manifolds, that is, pseudo-Riemannian manifolds $(M^n,g)$ admitting a null parallel distribution $\D$ of rank $r\leq\frac{n}{2}$. We show that $\D$ always integrates to a $G$-Lie foliation $\F_\D$, where $G$ is the simply connected Lie group with Lie algebra equal to the structure algebra $\g_\D$ of $\D$. The transverse holonomy group of $(M,g)$ coincides with the image of the holonomy morphism $h:\pi_1(M)\to G$. We prove that $\mathrm{Ric}(X,\cdot)=0$ for all $X\in\Gamma(\D)$, and show that in dimension~$3$ the model group is always $\R$, while in dimension~$4$ with rank~$2$ the structure algebra is always abelian. A local classification distinguishes the abelian, nilpotent, and solvable cases, and a rigidity theorem shows that a minimal nilpotent Walker foliation of dimension~$4$ cannot be deformed into a non-nilpotent solvable one.