- METRIC SPACES AND LENGTH SPACES 399
it in [18]). However, in general, for simply-connected complete noncom-
pact manifolds with nonpositive curvature, there is the notion of the ideal
boundary or sphere at infinity (see Ballmann, Gromov, and Schroeder [9]).
If (Mn, g) is a complete noncom pact Riemannian manifold with nonneg-
ative sectional curvature, then the asymptotic cone exists (see §6 of Appen-
dix I). On the other hand, there exist complete noncom pact Riemannian
manifolds with positive Ricci curvature which do not have asymptotic cones
(see Perelman's [150]). More elementarily, we have the following analogue
of Exercise G.24.
EXERCISE G.28. Give an example of a connected noncompact metric
subspace (X, d) of JE^2 and a point p E X such that there exist sequences
ai --7 0 and f3i --7 0 such that the Gromov-Hausdorff limits
.lim (X, aid,p) and .lim (X, f3id,p)
i--too i--+oo
both exist but are not isometric.
1.3.3. Euclidean metric cone.Let X be a topological space. The topological cone Cone (X) over Xis
the quotient of X x [O, oo) where the subset Xx {O} is identified to a point,
called the vertex. For example, Cone (sn-l) ~ }Rn (homeomorphic) for
n EN, where sn-l denotes the unit (n - 1)-sphere. Note that Cone (X) is
homeomorphic to a topological n-manifold if and only if Xis homeomorphic
to sn-l (in which case Cone (X) ~ IRn).
Given x EX and r E [O,oo), let [(x,r)] denote the equivalence class in
Cone (X) of (x, r).
DEFINITION G.29 (Euclidean metric cone). The Euclidean metric
cone of a metric space (X, d) with diam (X, d) ::::; 7f is Cone (X) with the
metric
(G.12) dcone(X) ([(x1, rl)] , [(x2, r2)]) = .J rr + r~ -2r1r2 cos (d (x1, x2))
defined for all x 1 , x2 E X and r1, r2 E [O, oo ). We call dcone(X) the cone
metric. For the definition of the Euclidean metric cone and its properties
when we have diam (X, d) > 7f, seep. 93 ff. in [18].
Observe that for any c E (0, oo)dcone(X) ([(x1, cr1)], [(x2, cr2)]) = cdcone(X) ([(x1, rl)], [(x2, r2)]).
Hence the map c: Cone (X) --7 Cone (X) defined by c ([(x, r)]) = [(x, er)] is
a homothety, i.e., for any y,z E Cone(X) we have dcone(X) (c(y) ,c(z)) =
cdcone(X) (y, z).
EXERCISE G.30. Show that indeed (Cone (X), dcone(X)) is a metric
space. See the proof of Proposition 3.6.13 in [18].The following remark gives one motivation for the definition in (G.12).