In this paper,we study a new Finslerian quantity■defined by the T-curvature and the angular metric tensor.We show that the■-curvature not only gives a measure of the failure of a Finsler metric to be of scalar flag ...In this paper,we study a new Finslerian quantity■defined by the T-curvature and the angular metric tensor.We show that the■-curvature not only gives a measure of the failure of a Finsler metric to be of scalar flag curvature but also has a vanishing trace.We find that the■-curvature is closely related to the Riemann curvature,the Matsumoto torsion and theΘ-curvature.We solve Z.Shen’s open problem in terms of the■-curvature.Finally,we give a global rigidity result for Finsler metrics of negative Ricci curvature on a compact manifold via the■-curvature,generalizing a theorem previously only known in the case of negatively curved Finsler metrics of scalar flag curvature.展开更多
基金supported by National Natural Science Foundation of China(Grant Nos.11771020 and 12171005)。
文摘In this paper,we study a new Finslerian quantity■defined by the T-curvature and the angular metric tensor.We show that the■-curvature not only gives a measure of the failure of a Finsler metric to be of scalar flag curvature but also has a vanishing trace.We find that the■-curvature is closely related to the Riemann curvature,the Matsumoto torsion and theΘ-curvature.We solve Z.Shen’s open problem in terms of the■-curvature.Finally,we give a global rigidity result for Finsler metrics of negative Ricci curvature on a compact manifold via the■-curvature,generalizing a theorem previously only known in the case of negatively curved Finsler metrics of scalar flag curvature.
