On unitary invariant strongly pseudoconvex complex Finsler metrics
On unitary invariant strongly pseudoconvex complex Finsler metrics
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DOI：
10.1016/j.difgeo.2015.02.002
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发表时间：
2015-06
影响因子：
0.5
通讯作者：
Chunping Zhong
中科院分区：
文献类型：
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作者：
Chunping Zhong
我们考虑一类形式为F=rϕ(t，S)的复Finsler度量，其中r=‖v‖2，t=‖z‖2，S=|<；z，v>；|2 r，其中z位于区域D⊂Cn和v∈Tz1，0 D中。我们证明了F是复Berwald度量当且仅当它来自Hermitian度量；F是Kähler Finsler度量当且仅当它来自Kähler度量。得到了F是弱复Berwald度量和弱Kähler Finsler度量的充要条件。我们的结果表明，存在大量具有酉不变性的弱复Berwald度量。证明了模为正常数的强凸复Finsler度量F是局部射影平坦或对偶平坦的当且仅当F是复欧氏度量。
We consider a class of complex Finsler metrics of the form F= r ϕ (t, s) with r=‖ v‖ 2, t=‖ z‖ 2 and s=|< z, v>| 2 r for z in a domain D⊂ C n and v∈ T z 1, 0 D. Complex Finsler metrics of this form are unitary invariant. We prove that F is a complex Berwald metric if and only if it comes from a Hermitian metric; F is a Kähler Finsler metric if and only if it comes from a Kähler metric. We obtain the necessary and sufficient condition for F to be weakly complex Berwald metrics and weakly Kähler Finsler metrics, respectively. Our results show that there are lots of weakly complex Berwald metrics which are unitary invariant. We also prove that, module a positive constant, a strongly convex complex Finsler metric F is locally projectively flat or dually flat if and only if F is the complex Euclidean metric.