1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. METRIC SPACES AND LENGTH SPACES 391

(see Exercise G.6 below). If there are no paths in A between x and y, then

the distance is defined to be oo.

Note that if£, : A ---+ IR is finite-valued, then de (x, y) < oo for any

(x, y) E Tc, where

Tc~ {(x, y) : 31 EA with/ (a)= x and/ (b) = y}.

DEFINITION G.4 (Length space). If the function [, of a quasi-length

space (X, I, A,£,) is nonnegative, invariant under reparametrizations, and

agrees with the topology of X, i.e., for every x EX and neighborhood U of

x

inf { [, ( rl[a,b]) : /EA with/ (a)= x and/ (b) ~ U} > 0,

then we say that (X, I, A,£,) is a length space. We call the function [,

a length structure; see §2.1 of [18] for more details. When I and A are

understood, we denote a length space simply by (X, £,).

Let (X, I, A,£,) be a length space. We say that a path a : I ---+ X

contained in A, where I C IR is an interval, is parametrized by arc length

if for all [ c, d] C I we have

(G.4) [, ( al[c,d]) = d - c.

LEMMA G.5. If f3 : I ---+ X is a rectifiable path in a length space, i.e.,

if £, (/3) < oo, then there exists a continuous nondecreasing function </> :

I ---+ [O, £, (/3)] and a rectifiable path /3 parametrized by arc length such that

f3 = j3 o ¢ (see Proposition 2.5.9 of [18]).

Let (X, £,) be a length space. A path I: [a, b] ---+Xis a shortest path
if for every path f3 joining/ (a) and/ (b) we have£, (/3) 2: £, (r). Note that
if/ : [a, b] ---+ X is a shortest path and if [c, d] C [a, b], then rl[c,d] is also
a shortest path. A path I : [a, b] ---+ X is a geodesic if for every T E [a, b]

there exists c: > 0 such that 11 [T-e,T+e]n[a,b] is a shortest path.

A length space (X, £,) is complete if every two points x, y EX can be

joined by a shortest path, i.e., there exists/ EA such that

£, (r) =de (x, y).

EXERCISE G.6 (Metric induced by a length space). Given a quasi-length

space (X,I,A,C), show that de satisfies the triangle inequality so that

(X, de) is a quasi-metric space. Moreover, if (X, I, A,£,) is a length space,
then (X, de) is a metric space (where de is allowed to take the value oo).
This is Exercise 2.1.2 of [18].

If a metric d is the associated metric of a length structure, then we say
that d is an intrinsic metric. If the length structure is complete, then we
say that d is a strictly intrinsic metric.