Why the gamma factors $\Gamma_{k l}^{j}$ are zero in flat submanifold?

Question

Axiom 2.1. The Newtonian spacetime is characterized geometrically by a pentuple $\mathbb{N}=\langle N, D, \Omega, V, \hat{h}\rangle$, where

1. $N$ is a paracompact, connected, oriented and noncompact fourdimensional smooth manifold.
2. $D$ is a linear connection on $N$ such that its tensors of torsion and curvature satisfy $$ T[D]=0 \quad \text { and } \quad R[D]=0 $$
3. $\Omega \in \sec \left(T^{*} N\right), \Omega \neq 0$, is a differentiable 1 -form field on $N$ such that $$ D \Omega=0 $$
4. $V \in \sec (T N)$ is a differentiable vector field on $N$ such that $\Omega(V)=1$
5. $\hat{h} \in T_{0}^{2} N$ is a two-covariant, symmetric, and differentiable tensor field on $N$ such that for every $p \in N$ (a) $\hat{h}_{p}\left(u_{p}, v_{p}\right)=0 \forall u_{p} \in T_{p} N \Leftrightarrow v_{p}=k V_{p}, k \in \mathbb{R}$ (b) $\hat{h}_{p}\left(u_{p}, u_{p}\right)>0 \forall u_{p} \in T_{p} N$ (c) $\left.D \hat{h}\right|_{p}=0$

Definition 2.2. Any function $t: N \rightarrow \mathbb{R}$ for which $\Omega_{p}=a d t_{p} \neq 0$ for all $p \in N, a \in \mathbb{R}, a>0$ , is called an (admissible) time function for $\mathbb{N}$. If $a=1$, the function $t$ is also said to be normalized. For each admissible time function $t: N \rightarrow \mathbb{R}$ , the number $t(p) \in \mathbb{R}$ is called time (relative to $t)$ of the event $p \in N$ and given two events $p, q \in N$, the number $|t(q)-t(p)|$ is called temporal interval (relative to $t$ ) between $p$ and $q$

Two events $p, q \in N$ are said to be simultaneous if and only if $t(p)=t(q)$. For each $p \in N$, the set $$ S_{p}=\{q \in N,t(q)=t(p)\} $$ of all events simultaneous with the event $p$ is called absolute simultaneity space at $p .\left(\right.$ Obviously,$S_{q}=S_{p}$ if $\left.t(p)=t(q) .\right)$

Proposition 2.4. For each $p \in N$ the set $S_{p}$ is a flat three-dimensional submanifold of $N$

Definition 3.1. A reference frame on $\mathbb{N}$ is characterized by a futurepointing timelike vector field $E \in \sec (T \mathscr{U}),\mathscr{U} \subseteq N$ , such that $$ \Omega_{p}\left(E_{p}\right)=1$$

Definition 3.3. We say that $E$ is inertial if and only if $$ D E=0 $$ at all points of $\mathscr{U}$.

Definition 3.22. A (proper) moving frame on $\mathbb{N}$, defined in $\mathscr{U} \subset N$, is a quadruple $\left\langle e_{\mu}\right\rangle=\left\langle e_{0}, e_{1}, e_{2},e_{3}\right\rangle, e_{\mu} \in \sec (T \mathscr{U}), \mu=0,1,2,3$ , of linearly independent differentiable vector fields on $\mathscr{U} \subset N$ such that, at each $p \in \mathscr{U}$,

1. $\Omega_{p}\left(e_{0}\right)=1$,
2. $\Omega_{p}\left(e_{k}\right)=0, k=1,2,3$. where $\Omega$ is a one form. The dual frame is $\left\langle\theta^{\mu}\right\rangle=\left\langle \theta^{0}=\Omega, \theta^{1}, \theta^{2}, \theta^{3}\right\rangle$

Definition 3.25. A moving frame $\left\langle e_{\mu}\right\rangle$ on $\mathscr{U} \subset N$ will be called coordinate if and only if, at each $p \in \mathscr{U}$, $$ \left.\mathscr{L}_{e_{\mu}} e_{v}\right|_{p}=0 $$ where $\mathscr{L}$ stands for the Lie derivative.

Proposition 3.33. A moving frame $\left\langle e_{\mu}\right\rangle$ on $\mathbb{N}$, defined in $\mathbb{U} \subset N$ , is coordinate if and only if $$ \Gamma_{\mu v}^{\rho}=\Gamma_{v\mu}^{\rho} $$

Definition 3.34. Let $E \in \sec (T \mathscr{}), \mathbb{U} \subseteq N$, be a reference frame field. A moving frame $\left\langle e_{\mu}\right\rangle$ on $\mathbb{N}$, defined in $\mathscr{U} \subset \mathbb{N}$ will be said to be naturally adapted to $E$ in $\mathscr{U}$ if and only if $$ e_{0 p}=E_{p} $$ Proposition 3.36. A moving frame $\left\langle e_{\mu}\right\rangle$ naturally adapted to an inertial reference frame $I \in \sec (T \mathscr{U}), \mathscr{U} \subseteq N$ is coordinate if and only if $$ \Gamma_{\mu v}^{\rho}=0 $$ in all points of $\mathscr{U}$, where $\Gamma_{\mu v}^{\rho}=\theta^{\rho}\left(D_{e_{\mu}} e_{v}\right)$

Proof: If $\Gamma_{\mu v}^{\rho}=0$ in $\mathscr{U}$, then obviously $\left\langle e_{\mu}\right\rangle$ is coordinate, since in this case $c_{\mu v}^{\rho}=\Gamma_{\mu v}^{\rho}-\Gamma_{v \mu}^{\rho}=0$ in $\mathscr{U}$.

Conversely, if $\left\langle e_{\mu}\right\rangle$ is coordinate, we have $\Gamma_{\mu v}^{\rho}=\Gamma_{v \mu}^{\rho}$ in $\mathscr{U}$ and therefore, taking into account that $e_{0}=I \Rightarrow D_{c_{\mu}} e_{0}=D_{e_{\mu}} I=0$ in $\mathscr{U}$, we have $$\Gamma_{\mu 0}^{p}=\Gamma_{0 \mu}^{\rho}=0 $$ in $\mathscr{U} .$ in $\mathscr{U}$. This concludes our proof, since the components $\Gamma_{k l}^{0}$ and $\Gamma_{k l}^{j}$ will be always null in $\mathbb{U}$, the first because $\Omega\left(e_{l}\right)=0$ and the other because the spacelike submanifolds of $N$ are flat.

The text in blockquote is from in this article The mathematical structure of Newtonian spacetime: Classical dynamics and gravitation .

I am not seeing why gamma factors $\Gamma_{k l}^{j}$ in preposition 3.36 are zero. Could anyone explain me why?

A contra example would be the basis associate to the spherical coordinates $(t,r,\theta,\phi)$ , $e_0=\frac{\partial}{\partial t},e_1=\frac{\partial}{\partial r}, e_2=\frac{\partial}{\partial \theta},e_3=\frac{\partial}{\partial \phi}$. They satisfy the condition above but not all the coefficient $\Gamma$ are zero.