On a Class of Weakly-Berwald $(\alpha ,\beta)$-Metrics |
Received:November 18, 2006 Revised:July 13, 2007 |
Key Words:
mean Berwald curvature weakly-Berwald metric $S$-curvature $(\alpha ,\beta)$-metric.
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Fund Project: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). |
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Abstract: |
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. |
Citation: |
DOI:10.3770/j.issn:1000-341X.2009.02.005 |
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