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| On Unitary Invariant Weakly Complex Berwald Metrics with Vanishing Holomorphic Curvature and Closed Geodesics |
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Citation： |
Hongchuan XIA,Chunping ZHONG.On Unitary Invariant Weakly Complex Berwald Metrics with Vanishing Holomorphic Curvature and Closed Geodesics[J].Chinese Annals of Mathematics B,2016,37(2):161~174 |
| Page view： 9390
Net amount： 4428 |
Authors： |
Hongchuan XIA; Chunping ZHONG |
Foundation： |
This work was supported by the National Natural Science
Foundation of China (Nos.11271304, 11171277), the Program for New
Century Excellent Talents in University (No.NCET-13-0510), the
Fujian Province Natural Science Funds for Distinguished Young
Scholars (No.2013J06001) and the Scientific Research Foundation
for the Returned Overseas Chinese Scholars, State Education
Ministry. |
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| Abstract： |
In this paper, the authors construct a class of unitary invariant
strongly pseudoconvex complex Finsler metrics which are of the form
$F=\sqrt{rf(s-t)}$, where $r=\|v\|^2,\ s=\frac{|\langle
z,v\rangle|^2}{r},\ t=\|z\|^2$, $f(w)$ is a real-valued smooth
positive function of $w\in\mathbb{R}$, and $z$ is in a unitary
invariant domain $M\subset\mathbb{C}^n$. 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
$F=\sqrt{rf(s-t)}$ on every Euclidean sphere
$\textbf{S}^{2n-1}\subset M$ are great circles. |
Keywords： |
Complex Finsler metrics, Weakly complex Berwald metrics, Closed
geodesics |
Classification： |
53C60, 53C40 |
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