On locally dually flat \((\alpha,\beta)\)-metrics with isotropic S-curvature
A Finsler metric \(F(x,y)\) is said to be locally dually flat if there exists a scalar function \(H(x,y)\), such that \[ G^i=\frac 12 g^{ij}H_{y^j}. \] The authors prove that: 1) a Finsler space with an \((\alpha,\beta)\) metric \[ F(x,y)=\alpha \phi \left(\frac{\alpha}{\beta}\right),\quad \phi'(0)=\phi'''(0)=0,\quad \phi'(0)\neq 0,\quad \frac{\phi''(0)}{\phi(0)}>-1\tag{*} \] is locally dually flat with isotropic \(S\)-curvature iff it is locally Minkowskian: 2) a locally dually flat Finsler space with \((*)\) and with isotropic \(S\)-curvature is locally projectively flat.
scientific article; zbMATH DE number 6150658
scientific article