290 7. THE REDUCED DISTANCE
Lengthri (;y)= fo'Tq J ~ + R + l'Y (7)J^2 d7
= lrq [f;J 1 + ~ ( R + J')t (7)J^2 )d7
= rq (N (l+N
7(R+J')t(7)J^2 ) +0(N-^2 ))d7
Jo V 2:;::
= forq [f;_d7 + forq ~ ( R + j')t (7)J^2 ) d7 + forq {f_o ( N-^312 ) d7
= fiNTq + k fo'Tq VT ( R + l'Y (7)1^2 ) d7 + ~o (N-^312 ).
The calculation indicates that as N -> oo, a shortest geodesic should ap-
proach a minimizer of the £-length functional defined by
.C('y) ~ forq VT (R('y(7) ,7) + J')t(7)J~(r)) d7.Note that the definition of .C('y) only depends on the data of (N, h).
EXERCISE 7 .4 (Levi-Ci vita connection of the potentially infinite metric).Consider the metric h on Nn x (0, T) defined in (7.16) by (7.12), (7.14),
and hiO = 0 (without the SN factor). The components of the Levi-Civita
connection N f(/ of h are defined by
Nfj a ~ = ~ Nfc ~
axa oxb 0 ab axe '
c=Owhere x^0 = 7. Show that
and