Holonomy on a curved sphere — geometric phase
Parallel transport of a vector along a closed loop on a sphere reveals holonomy — the vector rotates by an angle equal to the solid angle Ω enclosed by the loop. For a circle at polar angle θ, Ω = 2π(1 − cos θ). This geometric phase (Berry phase in quantum mechanics) is purely topological. The tangent vector (purple) returns rotated after completing the loop, even though it was "parallelly transported" at every step.