Each surface is a parametrization X(u,v). The tangent plane at a point is spanned by the exact partial derivatives X_u and X_v; the surface normal is their cross product. Gaussian curvature comes from the first and second fundamental forms, computed analytically from X_u, X_v and the second partials X_uu, X_uv, X_vv:
K = (LN − M²) / (EG − F²)
E=Xu·Xu F=Xu·Xv G=Xv·Xv
L=Xuu·n M=Xuv·n N=Xvv·n
Parallel transport carries a tangent vector around a closed loop by projecting it onto each successive tangent plane and rescaling to its original length — a discrete Levi-Civita transport. On a curved surface the vector returns rotated relative to its start; that rotation is holonomy. By the Gauss–Bonnet theorem the angle is approximately K times the area enclosed by the loop. The flat plane has K=0, so transported vectors never rotate — compare it against the sphere or saddle.