On unitary invariant weakly complex Berwald metrics with vanishing holomorphic curvature and closed geodesics

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• 作者：Hongchuan Xia ; Chunping Zhong
• 关键词：Complex Finsler metrics ; Weakly complex Berwald metrics ; Closed geodesics ; 53C60 ; 53C40
• 刊名：Chinese Annals of Mathematics - Series B
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：37
• 期：2
• 页码：161-174
• 全文大小：206 KB
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• 作者单位：Hongchuan Xia (1)
  Chunping Zhong (1)
  
  1. School of Mathematical Sciences, Xiamen University, Xiamen, Fujian, 361005, China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Applications of Mathematics
  Chinese Library of Science
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1860-6261

文摘

In this paper, the authors construct a class of unitary invariant strongly pseudoconvex complex Finsler metrics which are of the form \(F = \sqrt {rf\left( {s - t} \right)} \), where \(r = {\left\| v \right\|^2}\), \(s = \frac{{{{\left| {\left\langle {z,v} \right\rangle } \right|}^2}}}{r}\), \(t = {\left\| z \right\|^2}\), f(w) is a real-valued smooth positive function of w ∈ R, and z is in a unitary invariant domain M ⊂ C n . Complex Finsler metrics of this form are unitary invariant. We prove that F is a class of weakly complex Berwald metrics whose holomorphic curvature and Ricci scalar curvature vanish identically and are independent of the choice of the function f. Under initial value conditions on f and its derivative f′, we prove that all the real geodesics of \(F = \sqrt {rf\left( {s - t} \right)} \) on every Euclidean sphere S2n−1 ⊂ M are great circles. Keywords Complex Finsler metrics Weakly complex Berwald metrics Closed geodesics