412 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY
THEOREM G.60 (Regularity of Aleksandrov spaces). Let X be an n-
dimensional Aleksandrov space of curvature 2': k. Then we have the follow-
ing:
(1) There exists a set Xo C X such that X - Xo has n-dimensional
Hausdorff measure 0 and contains the set Sx of singular points of
x.
(2) The metric on Xis induced by a C^112 -Riemannian metric g on Xo.
(3) The Riemannian metric g extends continuously to X - Sx.
( 4) The space X has an almost everywhere approximately twice dif-
ferentiable structure in the sense of Stolz (see [146] for the
definition of such a structure; related to this is Definition H. 7 be-
low).
A geodesically complete Aleksandrov space of curvature 2': k is isomet-
ric to a C^1 •°' Riemannian metric on a smooth manifold (this is a result of
Nikolaev, with earlier work by Berestovskii; see Theorem 210 of [159]).
REMARK G.61. An important topic, which we do not discuss here, is
the behavior of the notions mentioned above under both Gromov-Hausdorff
convergence and collapsing of Aleksandrov spaces.
3. Notes and commentary
Advanced (and indispensable) references on metric geometry are the
original papers by Burago, Gromov, and Perelman [19] and Perelman [147].
Additional reference and survey articles include Grove [80], Shiohama [175],
Plaut [159], and Petrunin [156]. For a classic book on comparison geometry
for Riemannian manifolds, see Cheeger and Ebin [30]; more recent aspects
of comparison geometry are contained in the collection of papers edited by
Grove and Petersen· [82].
There are several excellent references for Aleksandrov spaces with cur-
vature bounded from below including the aforementioned [18], [19], and
Berestovskii and Nikolaev [12]. See also Fukaya [62], [159], and [156]. We
urge the reader to consult the aforementioned references.