Riemannian and K\"ahlerian Normal Coordinates

TL;DR

This paper compares K"ahler and Riemannian normal coordinates on K"ahler manifolds, providing a universal power series expansion of their difference based on curvature tensors, and extends these concepts to complex affine manifolds.

Contribution

It introduces a universal power series relating K"ahler and Riemannian normal coordinates and generalizes K"ahler normal coordinates to complex affine manifolds.

Findings

Derived explicit formulas for the difference between coordinate systems.

Developed an algorithm for calculating the power series to any order.

Provided formulas for Taylor series of geometric objects on symmetric spaces.

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.