= Circular curvature zeros for a planar SU2 exponential
{title2=$r=2\pi k$}
For $g=\exp(-i(x\sigma_1+y\sigma_2)/2)$, write $r=\sqrt{x^2+y^2}$ and $n=(x/r,y/r,0)$. The <Pauli matrix multiplication law> gives $g=\cos(r/2)I-i\sin(r/2)n\cdot\sigma$. On the circles $r=2\pi k$, the angular derivative is zero, so the two Cartesian right logarithmic derivatives are proportional to the same radial generator. They commute, giving zero <gauge curvature> for every scaling constant. At the origin their <commutator> is $-i\sigma_3/2$, which need not vanish.
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