Normal coordinates make the metric Euclidean throughout the chart

Differentiating the equation gives $T_q{S}^2=q^{\perp }$, consistently with the round metric in [F1]. Put . For smooth tangent vector fields $X,Y:S^2\to {\mathbb{R}}^3$, define ${\mathrm{\nabla }}_{X}Y=P(dY(X))$. The displayed formula is smooth and tangent-valued; pointwise linearity of $P$ and the ordinary directional-derivative product rule give real linearity in $Y$, function linearity in $X$, and ${\mathrm{\nabla }}_X(fY)=X(f)Y+f{\mathrm{\nabla }}_{X}Y$. Thus [F2] makes $\mathrm{\nabla }$ an affine connection. Moreover, for tangent $Y,Z$, projection does not alter the dot product with either field, so the ordinary dot-product rule gives $Xg(Y,Z)=g({\mathrm{\nabla }}_{X}Y,Z)+g(Y,{\mathrm{\nabla }}_{X}Z)\mathrm{.}$ Finally $dY(X)-dX(Y)=[X,Y]$ is tangent, whence ${\mathrm{\nabla }}_{X}Y-{\mathrm{\nabla }}_{Y}X=[X,Y]$. Thus the connection is metric compatible and torsion free, and uniqueness in [F2] identifies it with the round Levi--Civita connection.

For $v\in T_N{S}^2$ with $r=\mathrm{\mid }v\mathrm{\mid }>0$, put ${\gamma }_v(t)=\cos (tr)N+\sin (tr)v\mathrm{/}r$. Then $\mathrm{\mid }{\gamma }_v(t)\mathrm{\mid }=1$, ${\gamma }_v(0)=N$, ${\dot{\gamma }}_v(0)=v$, and ${\ddot{\gamma }}_v=-r^2{\gamma }_v$ is normal to the sphere. Applying the along-curve definition in [F2] to the projected connection of step 1.1 gives $D_t{\dot{\gamma }}_v=P_{{\gamma }_v}({\ddot{\gamma }}_v)=0$, so ${\gamma }_v$ is a geodesic. Its formula extends at $v=0$ by the constant curve, and uniqueness in [F3] yields ${\exp }_N(v)=\cos rN+(\sin r\mathrm{/}r)v$.

By [F3], restrict ${\exp }_N$ to a sufficiently small ball $B_{\rho }(0)\subseteq T_N{S}^2$ on which it is a diffeomorphism, and use the supplied basis to form normal coordinates. Choose $0<r<\min \{\rho ,\pi \mathrm{/}2\}$ and set $x=r{e}_1$, $w=e_2$. Since , differentiating the formula of step 2.1 in the direction $w$ gives $d({\exp }_N{)}_x(w)=(\sin r\mathrm{/}r)e_2$. This vector is exactly the second coordinate vector at $q={\exp }_N(x)$ because the inverse normal-coordinate chart is ${\exp }_N\circ E_e$. By [F4], $\sin r>\sin 0=0$; and the mean value theorem gives $c\in (0,r)$ with $\sin r-\sin 0=r\cos c$. Strict decrease of cosine on $[0,\pi ]$ gives $\cos c<\cos 0=1$, hence $0<\sin r<r$. Therefore the second coordinate vector has squared round length $g_{22}(q)={\left(\frac{\sin r}{r}\right)}^2<1.$

The metric coefficient in step 3.1 is not the Euclidean value $1$, even though [F3] gives $g_{22}(N)=1$ and vanishing first metric derivatives at the centre. This is the exact failure: normal coordinates normalize the metric's value and first derivatives at their centre, not its values throughout the chart. The witness is two-dimensional and uses an interior point with $r>0$; $r=0$ is precisely the normalized centre, while empty and zero-dimensional manifolds cannot supply this counterexample. Assumption [A1] is used only through the library interfaces collected in [F3]; every construction and calculation in steps 1.1--3.1 is explicit and makes no choice.