- ALEKSANDROV SPACES WITH CURVATURE BOUNDED FROM BELOW 407
A line in a length space Xis a (unit speed) path I: JR-+ X such that
rl[a,b] is a shortest path between/ (a) and/ (b) for any a, b ER
THEOREM G.45 (Aleksandrov space splitting theorem). If Xis an Alek-
sandrov space of nonnegative curvature which contains a line, then X is
isometric to the metric product of an Aleksandrov space of nonnegative cur-
vature with R
This splitting theorem generalizes the splitting theorem for Riemannian
manifolds with nonnegative sectional curvature. However, for Riemannian
manifolds, one only needs to assume that the Ricci curvatures are nonneg-
ative to prove a splitting theorem. As far as we know, the corresponding
generalization has not been achieved yet; related to this is the problem of
finding a good notion of nonnegative Ricci curvature on complete length
spaces. Note also that the notion of Busemann function, used in the proof
of the Riemannian version of the splitting theorem, can be generalized to
Aleksandrov spaces and is used in the proof of the above theorem. We shall
review Busemann functions on Riemannian manifolds in §1 of Appendix I.
The following is Theorem 10.4.1 of [18], which generalizes the corre-
sponding theorem for Riemannian manifolds.
THEOREM G.46 (Bonnet-Myers-type diameter bound). If Xis an Alek-
sandrov space of curvature ;:::: k, where k > 0, then we have
diam ( X) ::; 7r / vk.
REMARK G.47 (Volume comparison for Aleksandrov spaces). Another
basic technique of comparison geometry, namely the Bishop-Gromov volume
comparison theorem for Riemannian manifolds, also extends to Aleksandrov
spaces of curvature;:::: k (see §§10.6.2-10.6.3 of [18]).
In the next subsection we list some other important tools used in the
study of Aleksandrov spaces. We end this subsection by stating a well-
known open problem which asks how large the closure of the subcollection
of Riemannian manifolds with sectional curvature bounded from below in
the collection of Aleksandrov spaces of curvature bounded from below is (see
[108]).
PROBLEM G.48 (Aleksandrov spaces as Riemannian limits). Is every
finite n-dimensional Aleksandrov space of curvature ;:::: k isometric to the
limit of a sequence of complete Riemannian manifolds (whose dimensions
may be higher than n) with sectional curvatures ;:::: k' for some k' E JR?
Apparently, the consensus is that the answer to this question should be
'no'.
2.4. Tools for Aleksandrov spaces with curvature bounded from
below.
In this subsection we give a brief account of a few tools used in the study
of Aleksandrov spaces.