How many torsionless affine connections exist in general dimension?

Abstract

We study the question how many real analytic torsion-free affine connections exist locally on a smooth manifold M of dimension n. The families of torsion-free connections with skew-symmetric Ricci tensor and those with symmetric Ricci tensor are determined in terms of the number of arbitrary functions of n variables.

Keywords

• Affine connection
• Ricci tensor
• Cauchy-Kowalevski Theorem