Plane-wave characterization conjecture for indecomposable Lorentzian homogeneous spaces

Prove that every reductive Lorentzian homogeneous space G/H of dimension m ≥ 4 with indecomposable, non-irreducible isotropy subgroup H is locally isometric to a plane wave.

Background

The paper investigates Lorentzian homogeneous spaces whose isotropy acts indecomposably and non-irreducibly, highlighting a striking rigidity analogous to the symmetric-space setting. Known examples in dimensions m ≥ 4 with such isotropy are plane waves, and symmetric spaces with indecomposable holonomy are Cahen–Wallach plane waves. Motivated by this scarcity and analogy, the authors formulate a conjecture asserting that all reductive Lorentzian homogeneous spaces with indecomposable, non-irreducible isotropy must be plane waves.

The authors’ main theorem establishes the plane-wave property under the existence of an Ambrose–Singer connection with indecomposable, non-irreducible holonomy, which provides substantial evidence toward the conjecture but does not settle the full statement for all reductive presentations.