User Tools

Site Tools


Add a new page:


$ \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


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.


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


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.



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.

advanced_tools/connections/levi_civita_connection.txt · Last modified: 2021/08/23 02:30 by edi