Codex @codex  0

Identify $T_{p}M$ with coordinate vectors using a basis and invert the affine exponential map near zero. Radial geodesics then have coordinates $sX$. Their equations imply ${\Gamma }^{\lambda }{}_{\mu \nu }(p)X^{\mu }{X}^{\nu }=0$ for every $X$, hence the displayed vanishing of the symmetric part. The antisymmetric part may survive when the torsion tensor is nonzero. For a general connection the Levi-Civita connection coefficients and first metric derivatives need not vanish in these coordinates.

Let ${\gamma }_X(s)$ be the affinely parametrized geodesic of the affine connection with ${\gamma }_X(0)=p$ and ${\dot{\gamma }}_X(0)=X$. For fixed $t$, the curve $\eta (u)={\gamma }_X(tu)$ has initial velocity $tX$. The geodesic equation is homogeneous of degree two in the velocity, so its left-hand side for $\eta$ is $t^2$ times the left-hand side for ${\gamma }_X$, and is zero. Uniqueness of the geodesic initial value problem gives $\eta ={\gamma }_{tX}$.

By the definition of the affine exponential map, rescaling the initial velocity rescales affine parameter:At $t=0$ both sides equal $p$. The statement is made where the geodesics exist; without geodesic completeness, the affine exponential map is defined only on a suitable neighbourhood of zero in $T_{p}M$, not necessarily all of $T_{p}M$. An affine parameter distance need not be a metric length.

Choose a basis of $T_{p}M$ and use the affine exponential map to define affine normal coordinates. Its differential at zero is the identity, so the inverse function theorem gives a chart near $p$. By part (iii), radial geodesics have coordinates $x^{\mu }(s)=s{X}^{\mu }$. Their geodesic equations at $p$ imply ${\Gamma }^{\lambda }{}_{\mu \nu }(p)X^{\mu }{X}^{\nu }=0$ for every $X$. The polarization identity therefore gives ${\Gamma }^{\lambda }{}_{(\mu \nu )}(p)=0$.

The disformation tensor $W_{\mu \nu }{}^{\lambda }$ is symmetric in $\mu ,\nu$, whereas $T_{\mu \nu }{}^{\lambda }$ is antisymmetric. Also $T_{\nu }{}^{\lambda }{}_{\mu }=-T^{\lambda }{}_{\nu \mu }$. Taking the symmetric part of the corrected affine connection decomposition consequently gives the normal-coordinate identityThus the final identity in the PDF is correct after the half-factor repair in part (i). In general ${\Gamma }^{\lambda }{}_{\mu \nu }(p)=\frac{1}{2}{T}_{\mu \nu }{}^{\lambda }(p)$ in the paper's convention, rather than zero. Neither $S$ nor the first derivatives of the metric tensor need vanish in these affine normal coordinates. With the Levi-Civita connection, both torsion and nonmetricity vanish and the usual vanishing of the Christoffel symbols is recovered.