= Christoffel symbols vanish at the center of normal coordinates
{c}
{title2=$\Gamma^i_{jk}(p)=0$}
In <geodesic normal coordinates> at $p$, radial <geodesics> have coordinates $x(t)=tv$. The <geodesic equation> at $t=0$ gives $\Gamma^i_{jk}(p)v^jv^k=0$ for every $v$. The <Levi-Civita connection> is torsion free, making these coefficients symmetric in $j,k$; the <polarization identity> then gives $\Gamma^i_{jk}(p)=0$. This does not imply vanishing curvature, which depends on derivatives of the coefficients.
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