Similar articles

Abstract: In this work, we study a class of special Finsler metrics F called arctangent Finsler metric, which is a special (α,β)-metric, where α is a Riemannian metric and β is a 1-form. We obtain a sufficient and necessary condition that F is locally projectively flat if and only if α and β satisfy two special equations. Furthermore we give the non-trivial solutions for F to be locally projectively flat. Moreover, we prove that such projectively flat Finsler metrics with constant flag curvature must be locally Minkowskian.

[1] Bao, D., Robles, C., 2003. On Randers metrics of constant flag curvature. Reports on Mathematical Physics, 51(1):9-42.

[2] Bryant, R., 2002. Some remarks on Finsler manifolds with constant flag curvature. Houston J. Math., 28(2):221-262.

[3] Chern, S.S., Shen, Z., 2005. Riemann-Finsler Geometry. World Scientific, p.53.

[4] Hamel, G., 1903. Über die Geometrieen in denen die Geraden die kürzestensind. Math. Ann., 57(2):231-264.

[5] Mo, X., Shen, Z., Yang, C., 2006. Some constructions of projectively flat Finsler metric. Science in China-Series A: Mathematics, 49(5):703-714.

[6] Shen, Y.B., Zhao, L.L., 2006. Some projectively flat (α,β)-metrics. Science in China-Series A: Mathematics, 49(6):838-851.

[7] Shen, Z., 2003. Projectively flat Randers metrics of constant curvature. Math. Ann., 325(1):19-30.

[8] Shen, Z., 2004. Landsberg Curvature, S-Curvature and Riemann Curvature, in a Sampler of Riemann-Finsler Geometry. MSRI Series Vol. 50, Cambridge University Press, p.303-355.

[9] Shen, Z., 2006. On Some Projectively Flat Finsler Metrics. Http://www.math.iupui.edu/~zshen/Research/papers/ConstructionsOfProjectivelyFlatMetrics.pdf.

[10] Shen, Z., Civi Yildirim, G., 2005. On a class of projectively flat metrics of constant flag curvature. Canadian Journal of Math. (in Press). Http://www.math.iupui.edu/~zshen/Research/papers/ProjectivelyFlatMetricsShenYildirim.pdf.