A new quantity in Finsler geometry

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.