Riemannian and K\"ahlerian Normal Coordinates

Abstract

In every point of a K\"ahler manifold there exist special holomorphic coordinates well adapted to the underlying geometry. Comparing these K\"ahler 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 K\"ahler 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 K\"ahler normal potential and obtain explicit formulas for the Taylor series of all relevant geometric objects on symmetric spaces.