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Summary: A CONE SPLITTING THEOREM FOR ALEXANDROV SPACES
STEPHANIE B. ALEXANDER AND RICHARD L. BISHOP
Abstract. By a cone is meant a warped product IgF , where I is an interval
and the warping function g : I ! R0 is a solution of g 00 + Kg = 0, i.e.,
g 2 FK. These include metric products and linear cones (K = 0), hyperbolic,
parabolic, and elliptical cones (K < 0), and spherical suspensions (K > 0).
A cone over a geodesic metric space supports a natural K-aĆne function,
i.e., a function such that the restriction to every unit-speed geodesic is in FK.
Conversely, the main theorems of this paper show that on an Alexandrov space
X of curvature bounded below or above, the existence of a nonconstant K-
aĆne function f forces X to split as a cone (subject to a boundary condition
or geodesic completeness, respectively).
For K = 0 and curvature bounded below, X splits as a metric product with
a line; this case is due to Mashiko (2002). Some special cases for complete
Riemannian manifolds were discovered much earlier: by Obata (1962), for
K > 0, with the strong conclusion that X is a standard sphere; and by Innami
(1982), for K = 0. For K < 0, with the additional assumption that f has a
critical point, our theorem now gives the dual to Obata's theorem, namely, X
is hyperbolic space.
1. Introduction
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