Levi-Civita Connection in Local Orthonormal Frame

Theorem
Let $\struct {M, g}$ be a Riemannian 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 orthonormal 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:


 * $\Gamma^k_{ij} = \dfrac 1 2 \paren {c^k_{ij} - c^j_{ik} - c^i_{jk} }$