Definition:Connection Coefficients
Jump to navigation
Jump to search
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 $\nabla$ (with respect to the frame $\tuple {E_i}$).
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 4$: Connections. Connections in the Tangent Bundle