关于一类弱-Berwald的$(\alpha,\beta)$-度量

On a Class of Weakly-Berwald $(\alpha ,\beta)$-Metrics

中文关键词:  平均Berwald曲率  弱Berwald度量  $S$-曲率  $(\alpha,\beta)$-度量.

英文关键词:mean Berwald curvature  weakly-Berwald metric  $S$-curvature  $(\alpha ,\beta)$-metric.

中文摘要:

      本文研究了一类重要的定义在一个$n$-维流形上的$(\alpha,\beta)$-度量$F=(\alpha \beta)^{m 1}/\alpha^{m}$,其中$\alpha(y)=\sqrt{a_{ij}(x)y^iy^j}$为黎曼度量, $\beta(y)=b_i(x)y^i$为非零1-形式且$m$为不等于$-1$, $0$, $-1/n$的实数.得到了这类度量为弱-Belwald度量的充要条件.进一步,证明这类$(\alpha,\beta)$-度量具有迷向Belwald曲率当且仅当它具有迷向$S$-曲率.并且此时, $S$-曲率为0且该度量为弱-Belwald度量.

英文摘要:

      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.