On \(R\)-quadratic Finsler spaces (Q2714388)

In Finsler geometry there are several notions of curvature. For a Finsler space \((M,F),\) the Riemann curvature is a family of linear transformations \(R_y:T_xM\to T_xM\) where \(y\in T_xM\), with homogeneity \(R_{\lambda y}=\lambda^2 R_y, \forall\lambda>0\). If \(F\) is Riemannian, i.e., \( F(y)=\sqrt{g(y,y)}\) for some Riemannian metric \(g\), then \(R_y=R(.,y)y\), where \(R(u,v)\) denotes the Riemannian curvature of \(g\). NEWLINENEWLINENEWLINEA Finsler metric is said to be \(R\)-quadratic if its Riemann curvature \(R\) is quadratic in \(y\in T_x M\). NEWLINENEWLINENEWLINEThe main purpose of this paper is to prove the following: NEWLINENEWLINENEWLINETheorem. Let \((M,F)\) be a positively complete Finsler space with bounded Cartan torsion. Suppose that \(F\) is \(R\)-quadratic, then \(F\) must be a Landsberg metric. In particular every compact \(R\)-quadratic Finsler space must be Landsbergian. NEWLINENEWLINENEWLINEThis theorem tells us that for Finsler metrics on a compact manifold the following holds: NEWLINENEWLINENEWLINE\(\{\text{Berwald metrics}\} \subset\{ R\text{-quadratic metrics}\}\subset \{ \text{Landsberg metrics}\}\).NEWLINENEWLINENEWLINEFor a submanifold \(M\) in a Minkowski space \((V,F)\), the Cartan torsion must be bounded. NEWLINENEWLINENEWLINECorollary. For any positively complete submanifold \(M\) in a Minkowski space \((V,F)\), if the introduced Finsler metric \(F\) is quadratic, then \(F\) must be a Landsberg metric.NEWLINENEWLINENEWLINEThe structure equations are treated in the third part.