- LOCAL VERSUS GLOBAL GEOMETRY 299
2.2. Lifting the metric by the exponential map. Let re =re (p)
be the conjugate radius of p E Mn. Then
expP[B(O,rc) : B(O, re) -+Mn
is an immersion. Hence
g ~ ( expP[B(O,rc)) * g
is a smooth metric on B(O, re) C TpMn.
LEMMA B.35. There are no cut points of 0 in the open Riemannian
manifoldPROOF. Let d denote distance in B ( 0, re) measured with respect to the
metric g. It will suffice to show that d ( 0, V) = !VI for all V E B(O, re)· The
unit speed geodesics of ( B(O, re), g) emanating from 0 are the lines defined
for VE s;-^1 Mn c TpMn and t E [O, re) by iv: t 1---+ tV. We haveL9 ( ivl[o,11) = !VI,
and hence
J ( o, v) ~ !VI.To see that equality holds, let w : [O, a] -+ B(O, re) be any path with w (0) = 0
and w (a) = V. By Corollary B.2 of the Gauss lemma, we have\19r = (exp;lt (\lgf) = (exp;lL (:r) = R.
Since \il\
9= 1, it follows that
L9 (w) ~la lw (t)1 9 dt2:: la \w(t),R)
9dt= la (w(t),\19r) 9 dt
= - [r (w (t))] dt = r (w (a)) - r (w (0)) = r (V) = !VI.
l
ad
0 dt
Hence
J ( v, o) 2:: !VI.
D
Notice that the lemma impliesinj 9 (o) =re.
Applying Corollary B.21, we obtain the following conclusion.