A Rigidity Theorem for Convex Surfaces

Theorem 1. Let (S2, g) be the two dimensional sphere with a metric of class C1,1 whose Gaussian curvature satisfies 0 ≤ K ≤ 1. Then any simple closed geodesic γ on (S2, g) has length at least 2π. If the length of γ is 2π, then (S2, g) is isometric to the standard round sphere (S2, g0) and γ is a great circle on (S2, g0) or (S 2, g) is isometric to a circular cylinder of circumference 2π capped by two unit hemispheres and γ is a belt around the cylinder (see Figure 1). Thus if K is continuous (for example when g is at least C2) or if K > 0 then (S2, g) is isometric to the standard round sphere.