---
tag: CAZYGE1T4E
kind: concept
name: Levi-Civita connection
msc: [53B05, 53B20]
specializes: [JX65KSVHSJ]
sources:
  - url: https://en.wikipedia.org/wiki/Levi-Civita_connection
same_as:
  wikidata: Q548675
  wikipedia: Levi-Civita connection
---

## Summary

The Levi-Civita connection of a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] is the [[affine connection|JX65KSVHSJ]] that has no [[torsion|JZNQAN00ZJ]] and preserves the metric.
By the [[fundamental theorem of Riemannian geometry|78JRW8KTH8]] there is exactly one such connection.
It is the connection used to define geodesics, parallel transport and curvature of a Riemannian manifold.

## Explanations

### L1 human

On a curved surface there is no obvious way to compare vectors at different points, so differentiating a vector field is not obvious.
A connection is a rule that does this, and there are many such rules.
The Levi-Civita connection is the one rule that respects lengths and angles and does not twist.
On a surface in $\R^3$ it is what one gets by differentiating in $\R^3$ and then projecting onto the tangent plane.
For $\R^n$ with the usual metric it is ordinary differentiation of the components.