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Does anyone know if there is a formula to Christoffel Symbols in $\mathbb{R^4}$?

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It's the same as in any dimension: $$ \Gamma_{cab} = \frac{1}{2} \left(\frac{\partial g_{ca}}{\partial x^b} + \frac{\partial g_{cb}}{\partial x^a} - \frac{\partial g_{ab}}{\partial x^c} \right) = \frac{1}{2}\, \left(g_{ca, b} + g_{cb, a} - g_{ab, c}\right) = \frac{1}{2}\, \left(\partial_{b}g_{ca} + \partial_{a}g_{cb} - \partial_{c}g_{ab}\right). $$

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