Definition:Connection Coefficients

Definition
Let $M$ be a smooth manifold.

Let $\dim M$ be the dimension of $M$.

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

Let $TM$ be the tangent bundle of $M$.

For all $i \in \N_{>0} : i \le \dim M$ let $\tuple {E_i}$ be a smooth local frame for $TM$.

Let $\nabla$ be the connection on $M$.

For all $i, j, k \in \N_{> 0} : i, j, k \le \dim M$ let $\Gamma_{ij}^k : U \to \R$ be a smooth real function such that:


 * $\ds \nabla_{E_i} E_j = \Gamma^k_{ij} E_k$

where Einstein summation convention has been imposed.

Then the set of all $\Gamma^k_{ij}$ is known as the connection coefficients of $\nabla$ ( the frame $\tuple {E_i}$).