On a Class of Weakly-Berwald $(\alpha ,\beta)$-Metrics
Received:November 18, 2006  Revised:July 13, 2007
Key Word: mean Berwald curvature   weakly-Berwald metric   $S$-curvature   $(\alpha ,\beta)$-metric.
Fund ProjectL:the National Natural Science Foundation of China (No.10671214); the Natural Science Foundation of Chongqing Education Committee (No.KJ080620); the Science Foundation of Chongqing University of Arts and Sciences (No.Z2008SJ14).
 Author Name Affiliation XIANG Chun Huan School of Mathematics and Statistics, Chongqing University of Arts and Sciences, Chongqing 402160, China CHENG Xin Yue School of Mathematics and Physics, Chongqing Institute of Technology, Chongqing 400050, China
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In this paper, we study an important class of $(\alpha, \beta)$-metrics in the form $F=(\alpha \beta)^{m 1}/{\alpha^{m}}$ on an $n$-dimensional manifold and get the conditions for such metrics to be weakly-Berwald metrics, where $\alpha =\sqrt{a_{ij}(x)y^{i}y^{j}}$ is a Riemannian metric and $\beta=b_{i}(x)y^{i}$ is a $1$-form and $m$ is a real number with $m\not= -1, 0, -1/n$. Furthermore, we also prove that this kind of $(\alpha,\beta)$-metrics is of isotropic mean Berwald curvature if and only if it is of isotropic $S$-curvature. In this case, $S$-curvature vanishes and the metric is weakly-Berwald metric.