Direct Zeghib-style route to the plane-wave conjecture

Ascertain a direct proof strategy, within Zeghib’s geometric approach based on non-compact isotropy yielding totally geodesic null hypersurfaces, that proceeds from the existence of a G-invariant auto-parallel hyperplane distribution to a proof of the plane-wave characterization conjecture for Lorentzian homogeneous spaces with indecomposable, non-irreducible isotropy.

Background

Zeghib’s method shows that non-precompact isotropy in a Lorentzian homogeneous space yields totally geodesic null hypersurfaces, and for indecomposable, non-irreducible isotropy it produces a G-invariant auto-parallel hyperplane distribution whose leaves are totally geodesic. This is weaker than parallelism and does not immediately imply the plane-wave structure.

The authors note they do not know how to push this geometric framework further to obtain the plane-wave conclusion and therefore pursue Ambrose–Singer connections instead, leaving open the development of a direct Zeghib-style proof of the conjecture.