Asian Journal of Mathematics

Volume 24 (2020)

Number 3

Riemannian and Kählerian normal coordinates

Pages: 369 – 416



Tillmann Jentsch (Lehrstuhl für Geometrie, Institut für Geometrie und Topologie, Fachbereich Mathematik, Universität Stuttgart, Germany)

Gregor Weingart (Unidad Cuernavaca del Instituto de Matemáticas, Universidad Nacional Autónoma de México, Cuernavaca, Morelos, Mexico)


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.


Kähler potential, Spencer connection, hermitean symmetric spaces

2010 Mathematics Subject Classification

53C35, 53C55, 58A20

Received 8 June 2017

Accepted 27 June 2019

Published 9 October 2020