= Differential of the exponential map at zero
{title2=$(d\exp_p)_0=\operatorname{id}$}
For the <Riemannian exponential map> $\exp_p(v)=\gamma_v(1)$, uniqueness of the <geodesic equation> gives $\exp_p(tv)=\gamma_v(t)$. Differentiation at $t=0$ yields $(d\exp_p)_0v=v$. Smooth ODE dependence and the <inverse function theorem> therefore make $\exp_p$ a <diffeomorphism> on a neighborhood of zero, producing <geodesic normal coordinates>. This is a local assertion and requires no geodesic completeness.
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