Two-piece property in manifolds without focal points (Q1081133)

According to \textit{J. Bolton} [Q. J. Math., Oxf., II. Ser. 33, 159-167 (1982; Zbl 0445.53036)] tightness of submanifolds can be defined in ambient spaces without conjugate points. Similarly the spherical two- piece-property (STPP) can be defined using geodesic spheres and horospheres. The main result of this paper says that geodesic spheres and horospheres themselves have the STPP under the assumption that the manifold has no focal point (which implies that two asymptotic geodesic rays cannot meet).