Proof of Laugwitz Conjecture and Landsberg Unicorn Conjecture for Minkowski norms with $SO(k)\times SO(n-k)$-symmetry

Abstract

For a smooth strongly convex Minkowski norm $F:{\mathbb{R}}^n\to {\mathbb{R}}_{\ge 0}$, we study isometries of the Hessian metric corresponding to the function $E=\frac{1}{2}{F}^2$. Under the additional assumption that $F$ is invariant with respect to the standard action of $SO(k)\times SO(n-k)$, we prove a conjecture of Laugwitz stated in 1965. Further, we describe all isometries between such Hessian metrics, and prove Landsberg Unicorn Conjecture for Finsler manifolds of dimension $n\ge 3$ such that at every point the corresponding Minkowski norm has a linear $SO(k)\times SO(n-k)$-symmetry

A small flaw in the proof of Theorem 1.2, more precisely in the proof of Lemma 2.4 for dimension 3, was corrected

References in corpus (2)

• Finsleroid--Finsler Space with Berwald and Landsberg Conditions
• Applications of Nijenhuis geometry II: maximal pencils of multihamiltonian structures of hydrodynamic type