## Tohoku Mathematical Journal 2000 December SECOND SERIES VOL. 52, NO. 4

 Tohoku Math. J. 52 (2000), 533-553

Title NORMAL COORDINATE SYSTEMS FROM A VIEWPOINT OF REAL ANALYSIS

Author Norio Shimakura

Abstract. 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.