nLab
fundamental theorem of Riemannian geometry

Contents

Idea

The fundamental theorem of Riemannian geometry is the theorem which states that on a pseudo-Riemannian manifold there is a unique metric-compatible affine connection with vanishing torsion – the Levi-Civita connection.

In terms of integrability of G-structures this says that the torsion of an orthogonal structure vanishes, see at integrability of G-structures – Examples – Orthogonal structure.