关于一类弱-Berwald的$(\alpha,\beta)$-度量

In this paper, we study an important class of （α,β）-metrics in the form F = （α＋β）^m＋1/α^m on an n-dimensional manifold and get the conditions for such metrics to be weakly- Berwald metrics, where α = √aij（x）y^iy^j is a Riemannian metric and β = bi（x）y^i is a 1-form and m is a real number with m ≠ -1,0,-1/n. Furthermore, we also prove that this kind of （α,β）-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.

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

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.