- ALEKSANDROV SPACES WITH CURVATURE BOUNDED FROM BELOW 411
is concave in some neighborhood of x.
2.4.3. Quasi-geodesics.
Geodesics play a crucial role in Riemannian geometry. For Aleksandrov
spaces, the following generalization is useful.
DEFINITION G.56 (Quasi-geodesic). Let X be an Aleksandrov space of
curvature 2:: k without boundary. A path "( in X is called a quasi-geodesic
if for ).. E R and any >.-concave function f the function f o "( is >.-concave.
Note that a geodesic is a quasi-geodesic. Quasi-geodesics have nice prop-
erties such as being closed under limits and such that for any tangent vector
V there exists a quasi-geodesic emanating from its base and pointing in the
direction of V.
There are several other useful analytical tools in the study of Aleksan-
drov spaces: the directional derivative, the differential, and the gradient
curve of a function on an Aleksandrov space. In Appendix H we shall dis-
cuss some of these notions and ideas in a simpler setting: closed locally
convex subsets in Riemannian manifolds.
Another useful notion is that of extremal set, which can be used to give
a refinement of the notion of stratification of Aleksandrov spaces. Also note
that some of the results in Morse theory have been generalized by Perelman
to Aleksandrov spaces [149].
For all of these results, the reader may find more information either from
the surveys by Plaut [159] and Petrunin [156] or from the original papers
mentioned in the notes and commentary at the end of this appendix.
2.5. Advanced results on Aleksandrov spaces with curvature
bounded from below.
The following is Perelman's stability theorem (see Theorem 10.10.5
of [18] for the statement), whose proof is partly based on the so-called
deformation theorem of Siebenmann [17 4].
THEOREM G.57 (Perelman's stability theorem). Let X be a compact
n-dimensional Aleksandrov space of curvature 2:: k, where k E R There
exists c: > 0 such that if Y is a compact n-dimensional Aleksandrov space of
curvature 2:: k with dGH (X, Y) < c:, then Y is homeomorphic to X.
REMARK G.58. A recent detailed expository account of this theorem has
been given by V. Kapovitch [103].
As an immediate consequence, we have the following.
COROLLARY G.59 (Topology of small metric balls). Let X be as in the
theorem above. For every p E X there exists c: > 0 such that B (p, c:) is
homeomorphic to the tangent cone TpX.
The following result of Otsu and Shioya [146] tells us about the dif-
ferentiable structures of Aleksandrov spaces. (Perelman also studied the
Riemannian structure on a finite-dimensional Aleksandrov space; see [151].)