For $v$ near $0\in T_pM$, let $\gamma_v$ be the unique geodesic with $\gamma_v(0)=p$ and $\dot\gamma_v(0)=v$. The <exponential map> is $\exp_p(v)=\gamma_v(1)$.

The constant initial velocity $0$ gives $\exp_p(0)=p$. Varying the initial velocity through $sv$ yields $\exp_p(sv)=\gamma_v(s)$, so
$$
(d\exp_p)_0(v)=\left.\frac d{ds}\right|_{0}\exp_p(sv)=v.
$$
Thus $(d\exp_p)_0$ is the identity. The <inverse function theorem> makes $\exp_p$ a diffeomorphism from a neighborhood of $0$ onto a neighborhood of $p$. Coordinates from an orthonormal basis of $T_pM$ are the <geodesic normal coordinates>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 115 / 4 / b / Solution

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Choose an orthonormal basis of $T_pM$ and use its linear coordinates on a neighborhood of zero on which $\exp_p$ is a diffeomorphism. Transporting these coordinates through $\exp_p$ gives geodesic normal coordinates centered at $p$.


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 Incoming links: Geodesic normal coordinates