$ \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 [[advanced_tools:connections]] // The Levi-Civita connection is a mathematical tool that we use to [[advanced_tools:parallel_transport|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). ---- **Examples** The diagram below shows three concrete examples for connections (Christoffel symbols) on simple 2-dimensional manifolds. For a more detailed explanation see [[https://esackinger.wordpress.com/blog/lie-groups-and-their-representations/#metric_connect_curvature|Fun with Symmetry]]. {{:advanced_tools:metric_connect_curvature.jpg?nolink}} The motto in this section is: //the higher the level of abstraction, the better//. The Christoffel symbols appear in the most important equations of general relativity: the [[equations:einstein_equation|Einstein equation]] and the [[equations:geodesic_equation|geodesic equation]].