The length of a curve in a space of curvature ≤𝐾

Abstract

Let M be a compact ball in a Riemannian manifold with sectional curvatures ⩽ K \leqslant K . Suppose its radius R 0 {R_0} is less than the injectivity radius at the center of M and R 0 > π / 2 K {R_0} > \pi /2\sqrt K if K > 0 K > 0 . Denote by M 0 {M_0} a circle of radius R 0 {R_0} in the plane of constant curvature K and by κ \kappa the curvature of ∂ M 0 \partial {M_0} . Then any curve in M with curvature ⩽ χ > κ \leqslant \chi > \kappa is not longer than a circular arc of curvature χ \chi in M 0 {M_0} whose ends are opposite points of ∂ M 0 \partial {M_0} . Any curve in M with total curvature not exceeding some τ > 0 \tau > 0 ( τ = π / 2 \tau = \pi /2 if K ⩽ κ 2 K \leqslant {\kappa ^2} ) is not longer than the longest curve in M 0 {M_0} with the same total curvature whose tangent vector rotates in a permanent direction.