390 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY
If (X, dx) and (Y, dy) are metric spaces, then the product metric on
X x Y is defined by^5dxxY ((x1, Y1), (x2, Y2)) = V dx (x1, x2)^2 + dy (yi, Y2)^2.
We call the metric space (X x Y, dxxY) the metric product of (X, dx)
and (Y, dy).
A map f : (X, dx) -7 (Y, dy) between metric spaces is an isometry if
it is bijective and distance-preserving, i.e.,
dy (f (x1), f (x2)) = dx (x1, x2)
for all x1, x2 EX. In this case we say that (X, dx) and (Y, dy) are isomet-
ric. An injective distance-preserving map is called an isometric embed-
ding.
Given a metric space (Y, dy) and a map f : X -7 Y, the pullback of
the metric dy by f is defined by
(f*dy )(x1, x2) =:= dy (f (x1), f (x2))
for all x1, x2 EX.
EXERCISE G .3 (Pullback metric). Show that ( X, f* dy) is a pseudo-
metric space. Moreover, if f is injective, then (X, j*dy) is a metric space.
1.1.2. Length-type spaces.
There are different ways to define length spaces; here we present one ofthem. We say that a quadruple (X,'I, A,£), where Xis a topological space,
I C JR. is an interval, A is a set of continuous paths 'Y : [a, b] -7 X where
[a, b] c I, and
.C: A-7 lR.U {oo},
is a quasi-length space if A is closed under restrictions, concatenations,
and linear reparametrizations and .C is additive:^6
.C (a'-' (3) = .C (a)+ .C (f3)
and .C ( 'Yl[a,rJ) depends continuously on T. Here, if a
f3 : [b, c] -7 X satisfy a (b) = f3 (b ), where [a, c] c I, then
(a'-' (3) (u) =:= { ; ~:~
if u E [a, bJ,
if u E [b, c][a, b] -7 X anddenotes the concatenation of a and f3. The set A is called the class of
admissible paths and an element of it is called an admissible path.
If "( E A is such that .C ( 'Y) < oo, then we say that 'Y is rectifiable.
Given a Hausdorff quasi-length space (X,'I,A,.C), we may define the
associated quasi-metric (or associated quasi-distance) by
d.c (x, y) =:=inf{£("!) : 'YE A with 'Y (a)= x and 'Y (b) = y}
(^5) The notion of product clearly extends to pseudo-metric spaces.
(^6) By convention, a+ oo = oo +a= oo for a E JR?..