Properties of normal coordinates at the center

From [F1], $x(p)=E^{-1}(0_p)=0$. The inverse chart is $x^{-1}={\exp }_p\circ E$, so [F2] gives $d(x^{-1}{)}_0(e_i^{\mathrm{s}\mathrm{t}\mathrm{d}})=d({\exp }_p{)}_{0_p}(e_i)=e_i$; by the definition of coordinate tangent vectors this is ${\mathrm{\partial }}_i{\mathrm{\mid }}_p=e_i$. Orthonormality then gives $g_{ij}(p)=g_p(e_i,e_j)={\delta }_{ij}$.

If $v={\sum }_i{v}^i{e}_i$ and ${\gamma }_{p,v}(t)$ is in the normal neighbourhood, [F1] yields ${\gamma }_{p,v}(t)={\exp }_p(tv)$ and therefore $x({\gamma }_{p,v}(t))=E^{-1}(tv)=(t{v}^1,\dots ,t{v}^n)$. Thus every radial coordinate line is a geodesic on every connected parameter subinterval for which it remains in the chart.

Substitute the line from step 1.2 into the coordinate geodesic equation [F2] at $t=0$. Its second coordinate derivatives vanish, so for every $v\in {\mathbb{R}}^n$ and every $k$, ${\sum }_{i,j}{\mathrm{\Gamma }}^k{}_{ij}(p)v^i{v}^j=0$. Taking $v=e_i^{\mathrm{s}\mathrm{t}\mathrm{d}}$ gives ${\mathrm{\Gamma }}^k{}_{ii}(p)=0$. For $i\mathrm{\ne }j$, taking $v=e_i^{\mathrm{s}\mathrm{t}\mathrm{d}}+e_j^{\mathrm{s}\mathrm{t}\mathrm{d}}$ gives ${\mathrm{\Gamma }}^k{}_{ij}(p)+{\mathrm{\Gamma }}^k{}_{ji}(p)=0$; symmetry from [F3] makes both terms zero. Hence every Christoffel symbol vanishes at $p$.

Write ${\mathrm{\Gamma }}_{ij\mathrm{\ell }}={\sum }_m{g}_{m\mathrm{\ell }}{\mathrm{\Gamma }}^m{}_{ij}$. The two equations ${\mathrm{\Gamma }}_{kij}(p)=0$ and ${\mathrm{\Gamma }}_{kji}(p)=0$ from step 2.1 and [F3] read respectively ${\mathrm{\partial }}_k{g}_{ij}+{\mathrm{\partial }}_i{g}_{kj}-{\mathrm{\partial }}_j{g}_{ki}=0,{\mathrm{\partial }}_k{g}_{ji}+{\mathrm{\partial }}_j{g}_{ki}-{\mathrm{\partial }}_i{g}_{kj}=0$ at $p$. Adding and using $g_{ij}=g_{ji}$ gives $2{\mathrm{\partial }}_k{g}_{ij}(p)=0$, so every first metric derivative vanishes.

In dimension zero all indexed families and sums are empty, $x(p)=0$, and every assertion holds. In dimension one step 2.1 uses $v=1$ and gives the sole symbol and then the sole metric derivative as zero. On an empty manifold there is no centred chart. The zero vector gives the constant radial geodesic; chart-domain endpoints are excluded because the normal source is open, while every included parameter time is covered by step 1.2. The orthonormal basis is supplied, and ${\mathrm{A}\mathrm{C}}_{\omega }$ is inherited only through [F1]--[F2].