Fibrewise Real-Analytic Landsberg Metrics in Dimension at Least Four Are Berwald

Abstract

Let $M$ be a smooth manifold of dimension $n\ge4$, and let $F$ be a smooth, strongly convex Landsberg metric, regular on the entire slit tangent bundle. We prove that $F$ is Berwald if, at each fixed base point, the Minkowski norm is real-analytic on the entire slit tangent space. The proof rests on two analytic rigidity theorems for a single Minkowski space. The first asserts that, in dimension at least $3$, a real-analytic Minkowski norm whose Hessian metric is flat is induced by an inner product. The second asserts that, in dimension at least $4$, the Hessian metric of a real-analytic Minkowski norm is flat if the entire indicatrix carries a nonzero smooth symmetric quartic form satisfying a twisted Codazzi equation. The quartic form need not be real-analytic, and the second theorem is proved independently of the first. Applying these results to the angular Berwald quartic form shows that the metric is Riemannian on the open set of base points on whose fibres the Berwald curvature is not identically zero. Since every Riemannian metric is Berwald, this set must be empty. No analytic structure, compactness or completeness of the base manifold is assumed, and the metric need not be reversible.