Riemannian and Kählerian Normal Coordinates

2017/07/20 by Tillmann Jentsch, Jentsch, Tillmann, Gregor Weingart +1
Mathematics · #53C55 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C55

paper · pdf · doi:10.48550/arxiv.1707.06379

arxiv created 2017/07/20 · arxiv updated 2017/07/21

Abstract

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.

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