= Parallel transport around a spherical triangle
{title2=$\tau=\operatorname{Rot}(\alpha+\beta+\gamma-\pi)$}
For the positively oriented boundary of a <spherical triangle> on the unit sphere, <parallel transport> is rotation by its angle excess modulo $2\pi$. The tangent is parallel along each geodesic side and turns by $\pi-\alpha$ at a corner of angle $\alpha$. The three corner turns sum to $3\pi-\alpha-\beta-\gamma$, and combining them with parallel transport returns the initial tangent. Hence the transport angle is the negative of that sum modulo $2\pi$, giving the formula. The <spherical excess formula> identifies the angle with enclosed area; reversing traversal changes its sign.
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