Geodesic Flow on S²

Explore great circle geodesics on the 2-sphere, parallel transport of a tangent vector along a closed path, and the Gauss-Bonnet holonomy angle proportional to enclosed solid angle.

Latitude of circuit θ (°)30

Number of geodesic legs4

View elevation (°)25

Show great circles

Animation speed1.0

Holonomy: —

Gauss-Bonnet theorem: The holonomy angle φ acquired by parallel-transporting a vector around a closed loop = solid angle Ω enclosed by the loop.
For a circuit at latitude θ: Ω = 2π(1 − sin θ), so φ = 2π sin θ.
Parallel transport condition: The transported vector has zero covariant derivative along the path: ∇_γ̇ V = 0.
Connection to Berry phase: Holonomy IS geometric (Berry) phase for the sphere (adiabatic quantum mechanics ↔ spherical geometry).

Geodesic Flow on S² — Parallel Transport & Holonomy