强凸Kähler-Berwald度量的特征化
Characterization of strongly convex Kähler-Berwald metrics
AI总结：
本文研究了强凸Kähler-Berwald度量的几何特征化，证明了复结构与Cartan联络的关系，并给出了具有常数全纯截面曲率的度量的刚性定理。
摘要：
令$F: T^{1,0}M\rightarrow[0,+\infty)$是复流形$M$上的强凸复Finsler度量，$\pmb{J}$是复流形$T^{1,0}M$上的标准复结构。我们给出了强凸Kähler-Berwald度量的几何特征化。特别是，我们证明$\pmb{J}$相对于Cartan联络水平平行当且仅当$F$是Kähler-Berwald度量。我们还证明与$F$相关的Cartan联络和Chern-Finsler联络相等当且仅当$\pmb{J}$相对于Cartan联络既水平又垂直平行。基于这些结果，我们给出了具有常数全纯截面曲率的强凸Kähler-Berwald度量的刚性定理。
英文摘要：
Let $F: T^{1,0}M\rightarrow[0,+\infty)$ be a strongly convex complex Finsler metric on a complex manifold $M$ and $\pmb{J}$ the canonical complex structure on the complex manifold $T^{1,0}M$. We give a geometric characterization of strongly convex Kähler-Berwald metrics. In particular, we prove that $\pmb{J}$ is horizontally parallel with respect to the Cartan connection iff $F$ is a Kähler-Berwald metric. We also prove that the Cartan connection and the Chern-Finsler connection associated to $F$ coincide iff $\pmb{J}$ is both horizontal and vertical parallel with respect to the Cartan connection. Based on these results, we give a rigidity theorem of strongly convex Kähler-Berwald metrics with constant holomorphic sectional curvatures.
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