Point-normal and exponential coordinate charts of the metric-selected connection

(Point-normal chart.) Fix a smooth chart $(V,y)$ around $p$ with $y(p)=0$, and on a Euclidean ball around $0$ in $y(V)$ define $x^a:=y^a+\frac{1}{2}{\sum }_{b,c}{\mathrm{\Gamma }}^a{}_{bc}(p)y^b{y}^c$. Its derivative at $0$ is the identity matrix, so by [L3] $x$ restricts to a diffeomorphism from a neighborhood of $0$ onto an open set: it is a chart around $p$ with $x(p)=0$.

(Exponential chart.) Fix a $g_p$-orthonormal basis $e_0,\dots ,e_3$, let $\psi$ be its coordinate isomorphism, and let $W$ be a star-shaped neighborhood of $0$ on which ${\exp }_p$ is a diffeomorphism onto an open set $U_p$, as supplied by [L5]. Then ${\exp }_p(0)=p$, so $u:=\psi \circ {\exp }_p^{-1}$ is a smooth chart on $U_p$ with $u(p)=0$; it is the composition of the smooth maps ${\exp }_p^{-1}$ and the linear isomorphism $\psi$.

(Christoffel symbols vanish at $p$.) Differentiating the inverse relation $x(y(x))=x$ twice at $x=0$ where the first derivatives are the identity gives ${\mathrm{\partial }}^2{y}^a\mathrm{/}\mathrm{\partial }{x}^b\mathrm{\partial }{x}^c(0)=-{\mathrm{\partial }}^2{x}^a\mathrm{/}\mathrm{\partial }{y}^b\mathrm{\partial }{y}^c(0)=-{\mathrm{\Gamma }}^a{}_{bc}(p)$. Substituting this and the identity Jacobians into the transformation law of [L1], with $y$ as the old and $x$ as the new coordinates, gives ${\mathrm{\Gamma }}^x{}^a{}_{bc}(0)={\mathrm{\Gamma }}^a{}_{bc}(p)-{\mathrm{\Gamma }}^a{}_{bc}(p)=0$.

(Straight coordinate lines.) Let $k\in {\mathbb{R}}^4$ and put $v:={\psi }^{-1}(k)\in T_{p}M$; for $t$ in a neighborhood of $0$ with $tv\in W$, [L5] gives $u^{-1}(tk)={\exp }_p(tv)={\gamma }_{p,v}(t)$, which is a geodesic. Substituting $x(t)=tk$ into the coordinate geodesic equation of [L6] gives ${\sum }_{b,c}{\mathrm{\Gamma }}^a{}_{bc}(tk)k^b{k}^c=0$ for all $a$ and all $t$ in an interval, hence at $t=0$: ${\sum }_{b,c}{\mathrm{\Gamma }}^a{}_{bc}(0)k^b{k}^c=0$ for every $k\in {\mathbb{R}}^4$. First take $k=e_b$ for each $b$ to obtain ${\mathrm{\Gamma }}^a{}_{bb}(0)=0$. Given $a$ and two indices $b\mathrm{\ne }c$, apply this with $k=e_b+e_c$ and with $k=e_b-e_c$ and subtract to get $2({\mathrm{\Gamma }}^a{}_{bc}(0)+{\mathrm{\Gamma }}^a{}_{cb}(0))=0$, that is ${\mathrm{\Gamma }}^a{}_{(bc)}(0)=0$; torsion freedom gives ${\mathrm{\Gamma }}^a{}_{bc}(0)={\mathrm{\Gamma }}^a{}_{(bc)}(0)=0$.

(The radial portion of every geodesic through $p$ is a straight line.) Let $\gamma$ be a geodesic with $\gamma (t_0)=p$ and put $v_0:={\gamma }^{\mathrm{\prime }}(t_0)$. Translate the parameter so that $t_0=0$, and write the translated parameter again as $t$. For any $t$ in the domain of $\gamma$ with $t{v}_0\in W$, the scaling law of The geodesic initial-value problem and the exponential map of the metric-selected connection makes $t$ an allowed parameter of ${\gamma }_{p,v_0}$ and gives ${\gamma }_{p,v_0}(t)={\exp }_p(t{v}_0)$. By geodesic uniqueness in [L5], $\gamma (t)={\gamma }_{p,v_0}(t)$ there, so the inverse exponential chart returns exactly $t{v}_0\in W$ and $u(\gamma (t))=t\psi (v_0)$. Because $W$ is star-shaped, its intersection with the line $\mathbb{R}{v}_0$ is an interval containing $0$; this proves the assertion on the connected radial passage through $p$ and makes no claim about a later return to $U_p$ after that radial vector leaves $W$.

(Metric first derivatives vanish, normal form.) Applying the compatibility identity of [L2] in the chart $x$ at the point $p$ gives ${\mathrm{\partial }}_a{g}^x{}_{bc}(p)={\sum }_d({\mathrm{\Gamma }}^x{}^d{}_{ab}(p)g^x{}_{dc}(p)+{\mathrm{\Gamma }}^x{}^d{}_{ac}(p)g^x{}_{bd}(p))=0$ by step 2.1. Let $A$ be the matrix whose columns are the components in the chart $x$ of an orthonormal basis of $(T_{p}M,g_p)$, supplied by [L4], and set $z:=A^{-1}x$ near $p$. Under this constant linear coordinate change the transformation law of [L1] has no second-derivative term and gives ${\mathrm{\Gamma }}^z{}_{bc}(0)=0$; the metric components satisfy $g^z{}_{ab}(0)=g^x{}_{cd}(0)A^c{}_a{A}^d{}_b={\eta }_{ab}$ by the choice of $A$; and $\mathrm{\partial }{g}^z(0)=0$ because a constant linear change composes the derivative array with the fixed invertible Jacobians. This proves (a).

(Metric normal form in the exponential chart.) With ${\mathrm{\Gamma }}^u(0)=0$ from step 2.2, the compatibility identity of [L2] gives ${\mathrm{\partial }}_a{g}^u{}_{bc}(0)=0$. The coordinate basis at $p$ is ${\mathrm{\partial }}_{u^a}{\mathrm{\mid }}_p=d({\exp }_p{)}_0(e_a)\cdot$ in the identification of [L5], which equals $e_a$; hence $g^u{}_{ab}(0)=g_p(e_a,e_b)={\eta }_{ab}$.

Claims (a) and (b) are steps 3.1 and 1.2-3.2 respectively; the final assertion about second derivatives is a scope statement, and the proof above never differentiates the metric twice, so it asserts nothing at second order.