- ALEKSANDROV SPACES WITH CURVATURE BOUNDED FROM BELOW 409
by
expP (v, t) ~ x.(iii) The cut locus C (p) of p is the set of x E X such that any minimal
geodesic px is not properly contained in another minimal geodesic
starting from p. We define Gp C Op to consist of those equivalence
classes [ ( v, t)] with the property that there exists x E C (p) such
that v is in the direction of some px and t = d (p, x).Note that given a geodesic 'Y: [a, b] ---+ X in an Aleksandrov space, it is
not always possible to extend 'Y beyond 'Y (b), i.e., X may not be geodesi-
cally complete.
One application of the notion of the space of directions L;P is in defining
the boundary of an Aleksandrov space.
DEFINITION G.51 (Stratification). Let X be a topological space and let{Xi}! 1 be a collection of subsets of X. Then {Xi}! 1 is called a stratifi-
cation of X into topological manifolds if
(1) Xi are disjoint topological manifolds without boundary and
NLJxi =X,
i=l(2) dimX1 > dimX2 > · · · > dimXN,
(3) the set x: ~ U!k xi is closed in x for each k = 1, ... 'N.
We call the Xi strata of X. The following is part of Theorem 10.10.1
in [18].
THEOREM G.52 (Aleksandrov spaces admit stratifications). If X is an
n-dimensional Aleksandrov space of curvature ?: k, then X admits a strati-
fication into topological manifolds.
We can inductively define the boundary of an Aleksandrov space of
curvature ?: k. The boundary of a 1-dimensional Aleksandrov space is de-
fined to be its topological boundary. Now suppose that the boundary has
been defined for Aleksandrov spaces of curvature bounded from below and
dimension:::; n-1. We define the boundary of an n-dimensional Aleksandrov
space X of curvature bounded from below as the set of points p E X where
the space of directions L;P has nonempty boundary (here we used Theorem
G.49(3)).
2.4.2. The distance function and semi-concave functions on Aleksandrov
spaces with curvature bounded from below.
Given any metric space X, the most natural continuous function is the
distance function d ( ·, p) to a given point p E X. This function plays a
prominent role in comparison geometry and geometric analysis. It is well
known that even if the underlying space X is a smooth manifold, the distance