The length of a curve in a space of curvature $\leq K$

Abstract:

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