Torsion free means curvature free

First verify that the displayed $R$ has an intrinsic pointwise meaning. Expanding $R(fX,Y)Z$ by [F1] and [F2] leaves $fR(X,Y)Z-Y(f){\mathrm{\nabla }}_{X}Z+Y(f){\mathrm{\nabla }}_{X}Z=fR(X,Y)Z$. The definition is skew in $X,Y$, hence is also function-linear in $Y$. On replacing $Z$ by $fZ$, the two iterated derivatives contribute second-derivative coefficient $(X(Yf)-Y(Xf))Z$, which is canceled by the bracket derivative using [F3]. Their cross terms are $Y(f){\mathrm{\nabla }}_{X}Z+X(f){\mathrm{\nabla }}_{Y}Z$ in each order and cancel as well. What remains is $fR(X,Y)Z$. Additivity and real linearity follow by expansion. Thus $R$ is function-linear in all three inputs. In a local coordinate frame expand each input into its finite sum of components; these identities express $R$ as the corresponding trilinear sum of the smooth fields $R({\mathrm{\partial }}_i,{\mathrm{\partial }}_j){\mathrm{\partial }}_k$. Therefore its values depend only on pointwise inputs and its coefficients are smooth. The original expression uses intrinsic operations, so its vanishing is independent of coordinates.

Set $g=e^{2x^2}(d{x}^2+d{y}^2)$ on ${\mathbb{R}}^2$. This is smooth positive definite everywhere. Its only nonzero metric derivatives are ${\mathrm{\partial }}_x{g}_{xx}={\mathrm{\partial }}_x{g}_{yy}=4x{e}^{2x^2}$, while $g^{xx}=g^{yy}=e^{-2x^2}$ and off-diagonal entries vanish. Substitution in [F4] gives ${\mathrm{\Gamma }}^x{}_{xx}=2x$, ${\mathrm{\Gamma }}^x{}_{yy}=-2x$, ${\mathrm{\Gamma }}^y{}_{xy}={\mathrm{\Gamma }}^y{}_{yx}=2x$, with every other coefficient zero. The lower indices are symmetric, so torsion is zero by [F5].

Coordinate fields commute. From step 1.2, ${\mathrm{\nabla }}_{{\mathrm{\partial }}_x}{\mathrm{\nabla }}_{{\mathrm{\partial }}_y}{\mathrm{\partial }}_y={\mathrm{\nabla }}_{{\mathrm{\partial }}_x}(-2x{\mathrm{\partial }}_x)=(-2-4x^2){\mathrm{\partial }}_x$, whereas ${\mathrm{\nabla }}_{{\mathrm{\partial }}_y}{\mathrm{\nabla }}_{{\mathrm{\partial }}_x}{\mathrm{\partial }}_y={\mathrm{\nabla }}_{{\mathrm{\partial }}_y}(2x{\mathrm{\partial }}_y)=-4x^2{\mathrm{\partial }}_x$. Hence $R({\mathrm{\partial }}_x,{\mathrm{\partial }}_y){\mathrm{\partial }}_y=-2{\mathrm{\partial }}_x$, nonzero even at $x=0$, where every Christoffel symbol in step 1.2 is zero. By step 1.1 this is an intrinsic nonvanishing value, and step 1.2 supplies zero torsion. These explicit calculations refute the proposed implication without any later curvature, holonomy or surface-integration theorem.