Abstract. In this paper, we define a new projective invariant and call it W -curvature. We prove that a Finsler manifold with dimension n ≥ 3 is of constant flag curvature if and only if its W -curvature vanishes. Various kinds of projectively flatness of Finsler metrics and their equivalency on Riemannian metrics are also studied.M.S.C. 2010: 53B40, 53C60.

In this paper, we characterize locally dually flat and Antonelli m-th root Finsler metrics. Then, we show that every m-th root Finsler metric of isotropic mean Berwald curvature reduces to a weakly Berwald metric.

In this Note, we prove that every m-th root Finsler metric with isotropic Landsberg curvature reduces to a Landsberg metric. Then, we show that every m-th root metric with almost vanishing H-curvature has vanishing H-curvature.© 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. r é s u m éDans cette Note, nous montrons que toutes les métriques de Finsler racines m-ièmes ayant une courbure de Landsberg isotrope se réduisent à une métrique de Landsberg. Nous montrons ensuite que toutes les métriques de Finsler racines m-ièmes ayant une H-courbure presque nulle ont en fait une H-courbure nulle.

On isotropic Berwald metricsby Akbar Tayebi (Qom) and Behzad Najafi (Tehran)