On the scalar curvature of Kropina metrics. I (Q6551730)

The authors study Kropina metrics and their scalar curvature defined by \textit{H. Akbar-Zadeh} [Bull. Cl. Sci., V. Sér., Acad. R. Belg. 74, No. 10, 281--322 (1988; Zbl 0686.53020)]\N on a Finsler manifold. There are three main theorems in the paper:\N\N1. A Kropina metric \(F\) is of weakly isotropic scalar curvature with \N\[\Nr = n(n-1)\left( \dfrac{3\theta}{F} + \kappa \right) \tag{1}\N\]\Nif and only if \(F\) is an Einstein metric.\N\N2. Every Kropina metric of weakly isotropic scalar curvature satisfying (1) has vanishing \(\mathbf{S}_{BH}\)-curvature.\N\N3. If \(F\) is a Kropina metric with isotropic \(\mathbf{S}\)-curvature, then no non-constant conformal transformation makes it a metric of isotropic scalar curvature.\N\NAfter the introduction, Section 2 gives the preliminaries on Finsler manifolds with Kropina metrics. The definitions of Ricci curvature, Busemann-Hausdorff volume form and \(\mathbf{S}\)-curvature are given. \(\mathbf{S}_{BH}\) means the \(\mathbf{S}\)-curvature determined by the Busemann-Hausdorff volume form. Several propositions on Kropina metrics and Ricci curvature are also given with detailed proofs.\N\NIn Section 3, the scalar curvature of Kropina metrics is computed step by step. The result is complex and lengthy but in the weakly isotropic scalar curvature case becomes more lenient and is discussed further in Section 4, leading to the main results of this paper.\N\NIn Section 5, the Yamabe problem is considered for Kropina metrics with isotropic \(\mathbf{S}\)-curvature. The Yamabe problem for a Finsler manifold asks whether there exists a non-constant smooth real-valued function \(\sigma\) on \(M\) such that \(e^{\sigma}F\) is of isotropic constant curvature. The answer is negative, as stated in the third main result. Proofs are again detailed.