1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. METRIC SPACES AND LENGTH SPACES 397

Then 9J1 is precompact in the collection of metric spaces with respect to the

Gromov-Hausdorff distance.

The pointed version of this is the following (see Theorem 8.1.10 of [18]).

THEOREM G.19 (Criterion for precompactness -pointed version). Sup-

pose that a collection 9J1Pt of pointed metric spaces has the property that

for every E > 0 and p > 0 there exists N ( E, p) E N such that for every

(X, xo) E 9J1Pt, there exists an E-net S c B (xo, p) of B (xo, p) C X con-

sisting of at most N ( E, p) points. Then 9J1Pt is precompact in the collection

of pointed metric spaces with respect to the pointed Gromov-Hausdorff dis-

tance.

As an application of Gromov-Hausdorff distance, in the next subsection

we discuss another important notion associated with metric spaces.

1.3. Tangent cone and asymptotic cone of a metric space.

We shall discuss the tangent cones and asymptotic cones of a special

class of metric spaces.

DEFINITION G.20. We say that a metric space is boundedly compact

if every closed and bounded subset is compact.

1.3.1. Tangent cone of a boundedly compact metric space.

The tangent cone of a boundedly compact metric space (X, d) at a

point p EX is defined as the pointed Gromov-Hausdorff limit

(G.10)

provided this limit exists for any sequence { ak} --+ oo and is independent of
{ ak} (up to isometry).^10 The tangent cone, which is a metric space, reflects
the infinitesimal geometry at a point.
The tangent cone of an n-dimensional Riemannian manifold at any point
is isometric to Euclidean space JEn.^11 Given p E M and a local coordinate
system (U, {xi}) containing p, we have (U, a~g,p) c (M, a~g,p) is isomet-
ric to

(x (U), a~gij (x) dxidxj, x (p)) = (x (U), 9ij ( a-;;^1 x) dxidxj, x (p)) ,

where xi ~ akxi and we abused notation. The RHS limits to (TpM, g (p) , Op),
which is isometric to lEn.
The reader may wish to consider the following.
QUESTION G.21 (Tangent cone of a subset). Suppose that (X, d) is a
boundedly compact metric space and suppose that S C X is a boundedly
compact subset. Let p E S be such that both tangent cones (TpS, drPs) and

(^10) Later we abuse notation and call any sequential limit 'a tangent cone', where the
limits may be different for different sequences; see Exercise G.24.
(^11) Note that in normal coordinates {xi} we have 9ij (x) = Oij + 0 (lxl^2 ), that is,
l9ij (x) - Oij I ::::; C lxl^2 for some constant C < oo.