Weakly conformal Finsler geometry
In this paper, an extension of conformal equivalence for Finsler metrics is introduced and called weakly conformal equivalence and used to define weakly conformal transformations. The conformal Lichnerowicz-Obata conjecture is refined to weakly conformal Finsler geometry. It is proved that: If \(X\) is a weakly conformal complete vector field on a connected Finsler space \((M, F)\) of dimension \( n\geq 2 \), then, at least one of the following statements holds: (1) There exists a Finsler metric \( F_{1} \) weakly conformally equivalent to \(F\) such that \(X\) is a Killing vector field of the Finsler metric \( F_{1} \). (2) \(M\) is diffeomorphic to the sphere \(S^{n} \) and the Finsler metric is weakly conformally equivalent to the standard Riemannian metric on \( S^{n} \), and (3) \(M\) is diffeomorphic to the Euclidean space \(\mathbb R^{n} \) and the Finsler metric \(F\) is weakly conformally equivalent to a Minkowski metric on \(\mathbb R^{n} \). The vector field \(X\) with respect to the Minkowski metric is homothetic.NEWLINENEWLINENEWLINEIt is also proved that if the algebra of weak Killing vector fields of \(M\) has dimension more than \( \frac{n\left( n-1\right) }{2}+1 \) then, \(F\) is weakly isometric to a Riemannian metric of constant sectional curvature, where \(F\) is a Finsler metric on an \(n\)-dimensional manifold \( (n\neq 2,4) \).
scientific article; zbMATH DE number 6368941