= Connection components
{title2=$\Gamma^a{}_{bc}$}
= Connection coefficients
{synonym}
For a local <basis> $e_a$ of the <tangent bundle>, an <affine connection> has components defined by $\nabla_{e_c}e_b=\Gamma^a{}_{bc}e_a$. If $V=V^be_b$, then $(\nabla_{e_c}V)^a=e_c(V^a)+\Gamma^a{}_{bc}V^b$. The <connection components> depend on the chosen <basis> and have an inhomogeneous transformation law; they are not themselves a <tensor field>. In a <coordinate basis>, a <torsion-free connection> has symmetry in the two lower indices.
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