A note on symmetry of perpendicularity in a $G$-space with nonpositive curvature

Abstract

Let 9! be a G-space. We denote by m(a, b), for a, b£% a midpoint vSO that am(a,b)=-m{a,b)b ―abl2. The G-space 91 has nonpositive if every point p has a neighborhood S{p,jp), where 0<yp<pi(p) (see [3] for definition of pi(p)), such that for any three points a, b, c in S(p, jp) the relation 2m(a, b)m(a, c)<bc holds, and R has negative if 2m(a,b)m(a,c)<bc when a, b, c are not on one segment. Because a G-space 91 with nonpositive curvature is finite-dimensional according to V. N. Berestovskii [1], and hence 91 has domain invariance (see [4] p. 16), the universal covering space 'tit of 91is straight by Busemann [3] p. 254. Moreover the spheres in 91 are convex. The straight line L in a G-space is called a to the set M at /, if /eLflM and every point of L has/as a foot on M, i.e., qf=qM for any qeL. We say that perpendicularity between lines is symmetric if the following holds : if a straight line L is a perpendicular to a straight line G, then G is a perpendicular to L. We say that a set M of a G-space is totally convex if p,qzM implies that all geodesic curves from p to q are contained in M. If a closed set M of a G-space 91 in which the spheres are convex is totally convex, then for each pc'Sl there is a unique point qzM such that pq=pM. If the spheres of a straight G-space are convex, we denote by Wp the point set earring straight lines through peK(q, := {r; qr=o} but not through any point p' S(q,a) = {r; qr<a\, which are called the supporting lines of K{q, at p. K(q, is differentiable at peK(q, a) if no proper subset of the Wp decomposes the space.