Normal coordinate systems for pseudo-Riemannian metrics are investigated from a viewpoint of the theory of partial differential equations. Given a cartesian coordinate system $x$, a local metric for which $x$ is a normal coordinate system is determined by a metric tensor at the origin and any one of certain three matrix functions. These are related one another by three partial differential equations. Solvability of these equations in $C^{\infty}$ framework and power series expansion of solutions are discussed.
Tohoku Math. J. (2)
52(4):
533-553
(2000).
DOI: 10.2748/tmj/1178207754