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 $\text{D}$ of rank $r\le \frac{n}{2}$. We show that $\text{D}$ always integrates to a $G$-Lie foliation ${\text{F}}_{\text{D}}$, where $G$ is the simply connected Lie group with Lie algebra equal to the structure algebra ${\text{g}}_{\text{D}}$ of $\text{D}$. The transverse holonomy group of $(M,g)$ coincides with the image of the holonomy morphism $h:{\pi }_1(M)\to G$. We prove that $\mathrm{R}\mathrm{i}\mathrm{c}(X,\cdot )=0$ for all $X\in \mathrm{\Gamma }(\text{D})$, and show that in dimension~$3$ the model group is always $\mathbb{R}$, 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.