Cut locus in a Riemannian manifold

Abstract

We assume hereafter M satisfies the condition (B) at p, and assume sectional curvatures Kσ of M are bounded from below. Let i(p) be the injective radius at p, and let i(M) the injective radius of M. Then we can choose a Riemannian metric g on M as i(M)=1. We denote by L(γ) the length of a piecewise smooth curve γ. Let δ be a lower bound for sectional curvatures Kσ. Let K0=min {δ,0} and N(K0) a simply connected complete Riemannian manifold of constant sectional curvature K0. We consider a geodesic triangle abc in N(K0) which consists of