In this paper,we study a new Finslerian quantity(T)defined by the T-curvature and the angular metric tensor.We show that the(T)-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 T-curvature is closely related to the Riemann curvature,the Matsumoto torsion and the(Θ)-curvature.We solve Z.Shen's open problem in terms of the(T)-curvature.Finally,we give a global rigidity result for Finsler metrics of negative Ricci curvature on a compact manifold via the(T)-curvature,generalizing a theorem previously only known in the case of negatively curved Finsler metrics of scalar flag curvature.
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维普
