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Riemannian and Kählerian normal coordinates
Asian Journal of Mathematics ( IF 0.6 ) Pub Date : 2020-06-01 , DOI: 10.4310/ajm.2020.v24.n3.a1 Tillmann Jentsch 1 , Gregor Weingart 2
Asian Journal of Mathematics ( IF 0.6 ) Pub Date : 2020-06-01 , DOI: 10.4310/ajm.2020.v24.n3.a1 Tillmann Jentsch 1 , Gregor Weingart 2
Affiliation
In every point of a Kähler manifold there exist special holomorphic coordinates well adapted to the underlying geometry. Comparing these Kähler normal coordinates with the Riemannian normal coordinates defined via the exponential map we prove that their difference is a universal power series in the curvature tensor and its iterated covariant derivatives and devise an algorithm to calculate this power series to arbitrary order. As a byproduct we generalize Kähler normal coordinates to the class of complex affine manifolds with $(1, 1)$-curvature tensor. Moreover we describe the Spencer connection on the infinite order Taylor series of the Kähler normal potential and obtain explicit formulas for the Taylor series of all relevant geometric objects on symmetric spaces.
中文翻译:
黎曼和凯勒法线
在Kähler流形的每个点上,都存在特殊的全纯坐标,非常适合基础几何。将这些Kähler法线坐标与通过指数图定义的黎曼法线坐标进行比较,我们证明了它们的差是曲率张量及其迭代协变导数的通用幂级数,并设计了一种算法将该幂级数计算为任意阶。作为副产品,我们将Kähler法向坐标泛化为曲率张量为((1,1))的复杂仿射流形的类别。此外,我们描述了Kähler正势的无限阶泰勒级数上的Spencer连接,并获得了对称空间上所有相关几何对象的泰勒级数的显式公式。
更新日期:2020-06-01
中文翻译:
黎曼和凯勒法线
在Kähler流形的每个点上,都存在特殊的全纯坐标,非常适合基础几何。将这些Kähler法线坐标与通过指数图定义的黎曼法线坐标进行比较,我们证明了它们的差是曲率张量及其迭代协变导数的通用幂级数,并设计了一种算法将该幂级数计算为任意阶。作为副产品,我们将Kähler法向坐标泛化为曲率张量为((1,1))的复杂仿射流形的类别。此外,我们描述了Kähler正势的无限阶泰勒级数上的Spencer连接,并获得了对称空间上所有相关几何对象的泰勒级数的显式公式。