Commutativity of Flows and Vanishing of the Bracket Differential Geometry Minimal Geodesic in a Non-Trivial Homotopy Class Riemannian Geometry Bochner–Weitzenböck Formula Riemannian Geometry Second Variation Formula Riemannian Geometry Local Isometry from Complete Manifold is a Covering Map Riemannian Geometry Frobenius Integrability Theorem Differential Geometry Bonnet–Myers Diameter Theorem Riemannian Geometry Bochner Vanishing Theorem Riemannian Geometry