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$\Gamma_{ijk} \equiv -\bigl(\partial_{i}g_{jk}-\partial_{k}g_{ij}-\partial_{j}g_{ki}\bigr)$

# Levi-Civita Connection

also known as Christoffel Symbols; see also Connections

## Intuitive

The Levi-Civita connection is a mathematical tool that we use to parallel transport vectors around a manifold.

Parallel transport is just the simplest way to compare vectors at different points in the manifold.

Parallel is necessary, for example, to define the covariant derivative.

## Concrete

Christoffel symbols $\Gamma^i_{jk}$ are a particular type of connection that a Lorentzian manifold has (called the Levi-Civita connection).

Examples

The diagram below shows three concrete examples for connections (Christoffel symbols) on simple 2-dimensional manifolds. For a more detailed explanation see Fun with Symmetry.

## Abstract

The motto in this section is: the higher the level of abstraction, the better.

## Why is it interesting?

The Christoffel symbols appear in the most important equations of general relativity: the Einstein equation and the geodesic equation.