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$ \Gamma_{ijk} \equiv -\bigl(\partial_{i}g_{jk}-\partial_{k}g_{ij}-\partial_{j}g_{ki}\bigr)$
The Christoffel symbols are 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.
The Christoffel symbols appear in the most important equations of general relativity: the Einstein equation and the geodesic equation.