1.2. METRI C SPACE S AN D LENGT H SPACE S
3
The mai n poin t t o notic e fo r no w i s tha t ther e i s a certai n 'larg e scale ' spiri t
to Legendre' s argument . Instea d o f lookin g (i n th e spiri t o f differentia l geometry )
at 'ver y small ' triangle s an d thei r defect s th e ide a whic h lead s t o th e notio n o f
Gaussian curvature Legendre examines what happen s in the limit when distance s
become very large. I t is this idea that wil l be developed in this course. W e will study
the geometr y o f metric (an d other ) space s wit h a view t o retainin g th e large-scal e
structure an d dispensin g wit h irrelevan t small-scal e complications .
1.2. Metri c space s an d lengt h space s
As yo u know , a metri c spac e i s a se t X equippe d wit h a distanc e functio n
d: X x X R whic h i s positive , definite , symmetric , an d satisfie s th e triangl e
inequality:
d(x0x2) ^ d(x
0
xi) + d(x
1
x2).
It will be convenient sometime s to allow a metric to take the value +oo. Notic e that,
if w e mak e thi s convention , the n i t i s eas y t o se e tha t an y tw o point s o f a metri c
space tha t ar e a t distanc e +o o mus t belon g t o differen t connecte d components .
In Riemannia n geometry , th e distanc e betwee n tw o point s o f a (Riemannian )
manifold i s define d t o b e th e infimu m o f th e (Riemannian ) length s o f th e curve s
joining them . Suc h relation s betwee n th e metri c an d th e length s o f path s wil l b e
important fo r us. W e therefore defin e th e length of a path i n a general metric space.
DEFINITION
1.1 . Le t X be a metric spac e an d le t y: [0,1]— X be a continuou s
path. Th e length o f y i s the supremu m
£(
Y
)=sup{^d(Y(tO,Y(t
i + 1
))},
1
taken ove r al l partitions 0 = t o t i t ^ = 1 o f [0,1].
The lengt h i s a nonnegative rea l numbe r o r -foo .
DEFINITION
1.2. A (connected) metri c space X is a length space i f the distanc e
between an y tw o point s o f X is equa l t o th e infimu m o f th e length s o f th e curve s
joining them .
For example , an y connecte d Riemannia n manifol d M i s a length space . T o see
this, denot e by £(y) th e lengt h o f a path y a s defined i n 1.1above , and b y £R(Y ) it s
length a s defined i n Riemannian geometry . Th e Riemannia n lengt h £ R o f a curve is
greater tha n o r equa l to th e distanc e betwee n th e end-points , an d th e Riemannia n
length o f th e concatenatio n o f tw o curve s i s th e su m o f th e Riemannia n length s
of the curve s concatenated . I t no w follow s fro m th e definitio n (1.1 ) o f lengt h tha t
£(Y) ^ ^R(Y ) r a U Y- Sinc e b y constructio n th e distanc e betwee n tw o point s i s
the infimu m o f the Riemannia n length s o f the curve s joining them , w e see that M
is a length space .
A pat h y : [0,a] X is calle d a geodesic segment i f i t i s a n isometry o f th e
interval [0 , a] into X. (Not e that thi s is a stronger more global condition tha n
the 'locall y length-minimizing ' conditio n usua l i n differentia l geometry. ) I n partic -
ular, th e lengt h o f y i s equa l t o th e distanc e betwee n it s endpoints . Conversely ,
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