# Levi-Civita Connection in Local Frame

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## Theorem

Let $\struct {M, g}$ be a Riemannian or pseudo-Riemannian manifold with or without boundary.

Let $U \subseteq M$ be an open subset of $M$.

Let $\sqbrk {\cdot, \cdot}$ be the Lie bracket.

Let $\tuple {E_i}$ be a smooth local frame.

Let $c^k_{ij} : U \to \R$ be smooth functions such that:

- $\sqbrk {E_i, E_j} = c^k_{ij} E_k$

Let $\nabla$ is the Levi-Civita connection of $\struct {M, g}$.

Let $g$ be a Riemannian or pseudo-Riemannian metric, which locally reads:

- $g = g_{ij} \rd x^i \otimes \rd x^j$

where $\paren {g_{ij}}$ is a matrix of smooth functions.

Let $g^{ij}$ be the inverse of $\paren {g_{ij}}$.

Suppose $\Gamma^k_{ij}$ are the connection coefficients of $\nabla$.

Then in the frame $\tuple {E_i}$ we have:

- $\Gamma^k_{ij} = \dfrac 1 2 g^{kl} \paren {E_i g_{jl} + E_j g_{il} - E_l g_{ij} - g_{jm} c^m_{il} - g_{lm} c^m_{ji} + g_{im} c^m_{lj} }$

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## Proof

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## Source of Name

This entry was named for Tullio Levi-Civita.

## Sources

- 2018: John M. Lee:
*Introduction to Riemannian Manifolds*(2nd ed.) ... (previous) ... (next): $\S 5$: The Levi-Civita Connection. Symmetric Connections