Contemporary Mathematics
Volume 424, 2007
On an Extremal Property of Quadrilaterals in an
Aleksandrov Space of Curvature
~
K
I.
D. Berg and
I.
G. Nikolaev
Dedicated to Yurii' Reshetnyak on his 75th birthday
ABSTRACT. In this note we introduce an analog of the notion of the cosine
of the angle between two "directions", possibly based at different points of
a metric space. For two pairs of points, we introduce the notion of the K-
quadrilateral cosine,
cosqK;
in a space of constant curvature, it coincides with
the actual cosine of the angle between two tangent vectors under Levi-Civita
parallel translation. We prove that in an
!RK
domain of an Aleksandrov space
of curvature ::;
K,
we have lcosqKI ::; 1. Our principal result states: if,
for a quadrilateral with two non-adjacent "directed" sides of equal length in
an
!RK
domain, we have
cosqK =
-1 for those two sides, then the geodesic
convex hull of the quadrilateral is isometric to the geodesic convex hull of a
K-parallelogramoid in a two-dimensional space of constant curvature
K.
1.
Introduction
The distance between two tangent vectors to a Riemannian space, possibly
based at different points, is given by the Sasaki metric [Sl, S2]. A generalization
of the Sasaki distance to general metric spaces was given in [Nl, N2]. The starting
point of our definition of the Sasaki metric in an abstract metric space is the
---t
quadrilateral cosine. We will keep the notation
71
=
AB for an ordered pair (A, B)
in a metric space
(M,
p).
If
Q
=
{A, B, C,
D} is a quadruple of points of
M,
A
=f.
B, C
=f.
D,
then we define the quadrilateral cosine by
(AB GD)
= p2(A,
D)+
p2(C,
B)-
p2(A,
C)- p2(B, D)
cosq ' 2p(A, B)p(C, D) '
---t -----+
which equals the angle between vectors AB and CD in Euclidean space.
In a general metric space, the quadrilateral cosine can be greater than one. In
our paper [BeN], we have shown that the condition of
I
cosq
I
being not greater
than one is closely related to the nonpositiveness of the curvature of the metric
space in the sense of A. D. Aleksandrov. In particular, in an
!R0
domain of an
1991 Mathematics Subject Classification. Primary 53C20; Secondary 53C45, 53C24, 51K10.
Key words and phrases. Aleksandrov space of curvature::; K, K-quadrilateral cosine, planes
of curvature
K,
convex hull.
@2007 American Mathematical Society
http://dx.doi.org/10.1090/conm/424/08092
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