392 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY
1.1.3. Relations between metric spaces and length spaces.
Given a (quasi-)metric space (X, d) and an interval I c IR, we let A
denote the set of continuous paths 'Y: [a, b] -+ X, where [a, b] CI, and we
define .Cd : A -+ IR U { oo} by
(G.5) .Cd("!)= sup I:7= 1 d ("! h-1), 'Y h)),
Twhere T is a partition a =To ::::;; T1 ::::;; · · · ::::;; Tk = b of [a, b]. We call .Cd the
(quasi-)length structure induced by d.^7EXERCISE G.7 (Length induced by metric space). Showthat (X,I,A,.ld)
is a quasi-length space. Moreover, show that if (X, d) is a metric space, then
(X,I,A,.ld) is a length space (see pp. 34-35 of [18]).
Note that, by definition, for a length structure induced by a metric, all
continuous paths are admissible (but certainly not necessarily rectifiable).
Moreover, length structures induced by a metric are lower semi-continuous
(see Theorem 2.3.4(iv) of [18]).
Given a metric space (X, d), we have the induced intrinsic metric
defined by
(G.6)that is, the intrinsic metric is the metric associated to the length induced
by the original metric.
We have the following two facts.
- If dis an intrinsic metric, then d.cd = d (see Proposition 2.3.12 on p.
37 of [18]).
2. Given a lower semi-continuous length structure .l (with respect to
pointwise convergence), we have .ldc = .l (see Theorem 2.4.3 of [18]).
Given two length spaces (X,.Cx) and (Y,.ly), we define the product
length space as follows. Let d.cx and d.cy be the associated metrics of .lx
and .Cy, respectively. The product length structure .lxxY on X x Y is
the length structure induced by the product metric d.cx x d.cy (seep. 88 of
[18]).
Recall that a Riemannian metric defines at each point an infinitesimal
measure of length and angle. Integrating the infinitesimal measure of length
along paths, we obtain a length structure.
EXAMPLE G .8 (Riemannian manifold). Let (Mn, g) be a connected Rie-mannian manifold. Given a piecewise C^1 path 'Y : [a, b] -+ M, its length is
(^7) Given a partition T = {r;}~=o of [a, b], let Ld ('Yi T) ~ 2=~=
1 d ('Y (r;-1), 'Y (7.)). Note
that if T1 = { rl} :~a and T2 = { rl} :~a are partitions of [a, b], then the partition T1 U T2
satisfies