| 摘要: |
| 本文研究了强R-J不变Hermite流形的曲率结构问题.利用由度量一阶导数构成的张量T，建立了一般Hermite流形曲率分量$R_{\alpha\bar\beta\gamma\bar\delta}$的分解方法.
在强R-J不变条件下获得了完整曲率分解$R=R^{K}+S$（其中$R^{K}$为拟K\"ahler型主项，S为曲率修正项），并引入S-流形（满足$S\equiv0$），证明其曲率张量在局部坐标表达式上与Kaehler曲率完全一致，但非平凡S-流形不具备Kaehler曲率的对称性.进而推广了Kaehler流形的曲率研究框架，构成了一类比Kaehler流形更广泛且非平凡的几何对象. |
| 关键词: Kaehler几何 强R-J不变Hermite流形 曲率分解 S-流形 |
| DOI： |
| 分类号:O186.12 |
| 基金项目:国家自然科学基金（12201554） |
|
| CURVATURE DECOMPOSITION OF STRONGLY R-J INVARIANT HERMITIAN MANIFOLDS AND S-MANIFOLDS |
|
hujiaying
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| Abstract: |
| This paper studies the curvature structure of strongly R-J invariant Hermitian manifolds. By utilizing the tensor $T$ constructed from first-order derivatives of the metric, we establish a decomposition method for the curvature components $R_{\alpha\bar\beta\gamma\bar\delta}$ of general Hermitian manifolds. Under the strongly R-J invariant condition, we obtain the complete curvature decomposition $R=R^K+S$, where $R^K$ is the pseudo-Kaehler principal term and $S$ is the curvature correction term. We further introduce S-manifolds (satisfying $S\equiv 0$) and prove that their curvature tensors have local coordinate expressions completely identical to Kaehler curvature; however, non-trivial S-manifolds do not possess the symmetry of Kaehler curvature. Consequently, we extend the curvature research framework of Kaehler manifolds and construct a broader class of non-trivial geometric objects. |
| Key words: Kaehler geometry strongly R-J invariant Hermitian manifolds curvature decomposition S-manifolds |
