Differential of the exponential map at zero ($(d{\exp }_p{)}_0=\mathrm{id}$)

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 $(d{\exp }_p{)}_{0}v=v$. 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.