Riemannian Curvature: Parallel Transport
Transport a vector around a loop — the holonomy angle equals the integrated Gaussian curvature
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Gaussian curvature K
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∬K dA (Gauss-Bonnet)
Parallel transport: A vector V along curve γ satisfies ∇_γ'V = 0 (covariant derivative = 0). On curved surfaces, transporting around a closed loop rotates V by the holonomy angle.
Gauss-Bonnet theorem: The holonomy around a loop = ∬_Ω K dA (integral of Gaussian curvature over enclosed area).
Key cases: Sphere K=1/r² → holonomy = solid angle / r². Cylinder K=0 → no holonomy. Saddle K<0 → negative holonomy. Torus: ∬K dA = 0 (χ=0).