On a Class of Curvature Properties of Projectively flat Finsler (α,β) -Metric
A. R. Kavyashree *
Department of P.G. Studies and Research in Mathematics, Kuvempu University, Shankaraghatta - 577451,Shimoga, Karnataka, India.
Mallikarjun Y. Kumbar
Department of Mathematics, Mahantswamy Arts, Science and Commerce College, Haunsbhavi-581109, Haveri, Karnataka, India.
S. K. Narasimhamurthy
Department of P.G. Studies and Research in Mathematics, Kuvempu University, Shankaraghatta - 577451,Shimoga, Karnataka, India.
H. Anjan Kumar
Department of Mathematics, HMS Institute of Technology, Tumkur, Karnataka, India.
*Author to whom correspondence should be addressed.
Abstract
In this paper, we study a class of Finsler metric in the form \(F=\alpha+\beta+\frac{2 \beta^{2}}{\alpha}-\frac{\beta^{4}}{3 \alpha^{3}}\) where \(\alpha= \sqrt{a_{i j} y^{i} y^{j}}\) is a Riemannian metric, \(\beta=b_{i} y^{i}\) is a 1−form. We obtain a necessary and sufficient condition for \(F\) to be locally projectively flat. Further, we prove that such projectively flat Finsler metrics with the constant flag curvature must be locally Minkowskian.
Keywords: Finsler (α,β) –metrics, Flag curvature, Projectively flat, Riemannian metric