354 7. THE REDUCED DISTANCEwe have(^1) im-d. d{J ( ) p=1mpr l' 2 -d"( ~ d v ( pr=^2 -) v
p---tO p p---tO T
independent of 7'. Hence, by taking the limit of (7.133) as 1' ---+ 0, we have
'V~~o) <if = 0 and
dp p
pH Fm .Cexp (
1
;:;;:; V, p^2 1') = 1im (3 (p)
T--->O 2vr T--->O
is a constant speed geodesic with respect to g ( 0) and with initial vector V.
Thus when we evaluate it at p = 1, we get that the limit is expg(O) (V). D
Next we compute the differential of Lr exp.
LEMMA 7.92. For 1' E (0, T), .Cr exp is differentiable at V and the tan-
gent map
is given by
(.Cr exp)* (W) = J('F),
where J(r) is the £-Jacobi field along .Cexp(V,r) with
J(O) = 0 and -d d I J(-er2
4
) = W.
er a=O
PROOF. We have a family of £-geodesics .Cexp(V + sW, r), T E [O, 7'],
for s E (-c:, c:). By the definition of .C-J aco bi field,
J(r) ~ : I .Cexp(V + sW, r)
s s=O
is an £-Jacobi field. Since .C exp(V + sW, 0) = p, we have J(O) = 0. From
d
d I .Cexp(V + sW, r) = V + sW,
er T=O
where er= 2y7, we get by taking fa 1s=O,
d
d I J(r) = w
er T=O
Note that the tangent map of .Cr exp at V is given by
D [.Cr exp(V)] (W) = :s ls=O .Cexp(V + sW, 1').
The lemma follows. D
As a simple corollary of the lemma we have