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 Einstein summation convention has been imposed.

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

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

Then:

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

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 4$: Connections. Connections in the Tangent Bundle