A local classification of a class of (α,β) metrics with constant flag curvature
A local classification of a class of (α,β) metrics with constant flag curvature
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DOI：
10.1016/j.difgeo.2009.05.008
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发表时间：
2010-04
影响因子：
0.5
通讯作者：
Linfeng zhou
中科院分区：
文献类型：
--
作者：
Linfeng zhou
首先计算了（α，β）度量的黎曼曲率和Ricci曲率。然后应用这些公式讨论了一类特殊的（α，β）度量F=α（1+βα）p（|p| 1）具有恒定的旗曲率。得到了F=（α+β）2α具有常旗曲率的充要条件.然后我们证明了这样的度量必须是局部射影平坦的，并完成了它们的局部分类。用同样的方法，我们得到了F=α2α+β的旗曲率为常数的必要条件，并证明了F=α2α+β不存在非平凡的松本度量.此外，我们还给出了一个否定的答案，即是否存在非平凡度量F=α（1+βα）p（|p|当β闭合时，旗曲率为常数。
We first compute Riemannian curvature and Ricci curvature of (α,β) metrics. Then we apply these formulae to discuss a special class (α,β) metrics F=α(1+βα)p(|p|⩾1) which have constant flag curvature. We obtain the sufficient and necessary conditions that F=(α+β)2α have constant flag curvature. Then we prove that such metrics must be locally projectively flat and complete their local classification. Using the same method we find a necessary condition that flag curvature of F=α2α+β is constant and proved that there are no non-trivial Matsumoto metrics. Furthermore, we give a negative answer whether there are non-trivial metrics F=α(1+βα)p(|p|⩾1) of constant flag curvature when β is closed.