#
nLab

fundamental theorem of Riemannian geometry

### Context

#### 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*.