The Rigidity of the closed three dimensional regular Landsberg metrics

Abstract

The Landsberg Berwald conjecture asks whether every regular Landsberg metric is Berwald. We settle this conjecture for closed three-dimensional manifolds: every smooth strongly convex regular Landsberg metric on such a manifold is Berwald, without any reversibility assumption. Equivalently, there are no regular Landsberg "unicorns" on closed three-manifolds; this is a reformulation of the same conjecture, rather than a second independent conjecture.The proof combines a vanishing theorem for commuting Codazzi cubic tensors on closed surfaces, a rigidity theorem for three dimensional Minkowski norms with constant curvature indicatrices, and a rank one argument for the nonlinear curvature. The remaining R-quadratic case is settled by compactness along the geodesic flow. The fibrewise constant curvature theorem is an ingredient in this argument and does not assert the full Laugwitz conjecture.