The geometry of the tangent cone to a G-space of nonpositive curvature with distinguished family of segments

摘要

We present a construction of tangent cone to a Busemann G-space with distinguished family of segments with the additional condition of nonpositivity of curvature in the sense of Busemann with respect to the distinguished family. We prove that the tangent cone in question has geometric properties similar to those of the tangent cone of a standard G-space of nonpositive curvature. Earlier, using tangent cones, the first author proved H. Busemann’s conjecture for G-spaces of nonpositive curvature which states that each such a space is a topological manifold. The tangent cone constructed in this paper can be used as a basic tool in the generalization of this theorem to the class of spaces in question.