= Reflection holonomy along a closed projective geodesic
{title2=$\operatorname{diag}(1,-1)$}
In the round <real projective plane>, project the great-circle arc $(\cos t,\sin t,0)$ for $0\le t\le\pi$. Its endpoint and velocity match after antipodal identification, so it is a closed <geodesic>. On the sphere its tangent and the constant vector $(0,0,1)$ are parallel. Endpoint identification fixes the returned tangent and reverses the returned transverse vector. Its <parallel transport> is therefore a reflection. Nonorientability permits determinant-minus-one holonomy.
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