Curvature properties of \((\alpha,\beta)\)-metrics. (English) Zbl 1151.53019

Bácsó, Sándor; Cheng, Xinyue; Shen, Zhongmin

Sabau, Sorin V. (ed.) et al., Finsler geometry, Sapporo 2005. In memory of Makoto Matsumoto. Proceedings of the 40th Finsler symposium on Finsler geometry, Sapporo, Japan, September 6–10, 2005. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-42-6/hbk). Advanced Studies in Pure Mathematics 48, 73-110 (2007).

Let \(\alpha^2\) be a Riemannian metric where \(\alpha = \sqrt{a_{ij}(x)y^iy^j}\) and \(\beta = b_k(x)y^k\) is a 1-form on a manifold \(M^n\). The Finsler metric \(F=F(x, y)\) on \(M^n\) is called \((\alpha, \beta)\)-metric if \(F\) admits a presentation \(F(x, y) = f(\alpha, \beta)\) where \(f\) is homogeneous of degree 1. The paper is a survey of the recent developments in the curvature theory of Finsler \((\alpha, \beta)\)-metrics. The following questions are considered: volume forms and the computation of \(S\)-curvature, Randers metrics of scalar flag curvature, conditions for general \((\alpha, \beta)\)-metrics to be of Landsberg, of Douglas type and projectively flat.
For the entire collection see [Zbl 1130.53005].

Reviewer: P. D. Andreev (Arkhangelsk)

MSC:

53B40	Local differential geometry of Finsler spaces and generalizations (areal metrics)
53C60	Global differential geometry of Finsler spaces and generalizations (areal metrics)

Keywords:

Finsler manifold; \((\alpha, \beta)\)-metric; Randers metric; \(S\)-curvature; Busemann-Hausdorff volume form; Holmes-Thompson volume form; scalar flag curvature; Landsberg metric; Douglas metric