On a class of Finsler metrics of scalar flag curvature
Let \(M\) be a smooth manifold of dimension \(n\). A Finsler metric having the form \(F =\alpha\phi(b^2, s);\,\, s:= \frac{\beta}{\alpha}\), is called a general \((\alpha,\beta)\)-metric, where \(\alpha= \sqrt {a_{ij}(x)y^iy^j}\) is a Riemannian metric, \(\beta = b_i(x)y^i\) is a 1-form on \(M\) and \(b:= b(x) = ||\beta(x)||_\alpha\) is the norm of \(\beta\) with respect to \(\alpha\). In particular, if \(\phi_1 = 0\), the metric \(F\) reduces to the usual \((\alpha,\beta)\)-metric \(F = \alpha\phi(s)\). A Finsler metric \(F\) on \(M\) is said to be locally projectively flat if at any point of \(M\), there is a local coordinate system in which \(F\) is projectively flat. It is known that every locally projectively flat Finsler metric is of scalar flag curvature. However, the converse is not true. The author of the present paper is concerned with this converse. He tries to characterize Finsler metrics of scalar (resp. constant) flag curvature for a certain class of general \((\alpha,\beta)\)-metrics. He proves that a class of general \((\alpha,\beta)\)-metrics satisfying certain conditions is of scalar flag curvature if and only if it is locally projectively flat. Moreover, he finds a system of nonlinear partial differential equations characterizing the above class of general \((\alpha,\beta)\)-metrics such that it is of scalar (resp. constant) flag curvature and determines its local structure by solving these equations. As an application, the author obtains some non-projectively flat Finsler metrics of scalar (resp. constant) flag curvature.
scientific article; zbMATH DE number 6687990
scientific article