✎ Levi-Civita Connection [The Physics Travel Guide]

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

====== Levi-Civita Connection ======

//also known as Christoffel Symbols; see also [[advanced_tools:connections]] //

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

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

<tabbox Why is it interesting?> 
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]].

</tabbox>

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