Transformation Law for Connection Coefficients

Theorem
Let $M$ be a smooth manifold with or without boundary.

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

Let $\nabla$ be a connection in $TM$.

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

Let $\tuple {E_i}$ and $\tuple {\tilde E_j}$ be smooth local frames for $TM$ on $U$.

Suppose $\tuple {E_i}$ and $\tuple {\tilde E_j}$ are related by:


 * $\tilde E_i = A^j_i E_j$

where $\paren {A^j_i}$ is a matrix of functions.

Let $\Gamma^k_{ij}$ and $\tilde \Gamma^k_{ij}$ denote connection coefficients $\tuple {E_i}$ and $\tuple {\tilde E_i}$.

Then:


 * $\ds \tilde \Gamma^k_{ij} = \paren{A^{-1}}^k_p A^q_i A^r_j \Gamma^p_{qr} + \paren{A^{-1}}^k_p A^q_i \map {E_q} {A^p_j}$