= Affine normal coordinates
{title2=$\Gamma^\lambda{}_{(\mu\nu)}(p)=0$}
Identify $T_pM$ with coordinate vectors using a basis and invert the <affine exponential map> near zero. Radial <geodesics> then have coordinates $sX$. Their equations imply $\Gamma^\lambda{}_{\mu\nu}(p)X^\mu X^\nu=0$ for every $X$, hence the displayed vanishing of the symmetric part. The antisymmetric part may survive when the <torsion tensor> is nonzero. For a general connection the <Levi-Civita connection> coefficients and first metric derivatives need not vanish in these coordinates.
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