8. .C-JACOBI FIELDS AND THE .C-EXPONENTIAL MAP 347Thus we have a linear second-order ODE for the .C-J acobi field Y ( r) , called
the C-J acobi equation:(7.121)1
\lx (\lxY) = R (X, Y) X + "2\ly (\l R) - 2 (\ly Re) (X)
1- 2 Re (\l x Y) - 2 r \l x Y.
Since r = 0 is a singular point because of the ~ factor in the last term,
we rewrite the equation asDv-rx (\l v-rxY) = r ( \lx (\lxY) + 2 ~ \lxY)
= R (vfTX, Y) y!TX + ~\ly (\JR)
- 2y!T (\ly Re) ( y!TX) - 2ylTRc ( \l v-rxY).
Let Z(!7) ~ .jTX(r), where O" = 2./i and {3 (!7) = '""( (!7^2 /4). Then Z(!7) =
~~ and we can rewrite the .C-Jacobi equation for Y (r) as\l z (\l z Y) = -2!7 Re (\l z Y) + R ( Z, Y) Z
(7.122)(72- 2!7 (\ly Re) (Z) +
2
\ly (\l R),
where we view Y ( !7^2 / 4) as a function of O". Suppose Z ( 0) = lim 7 _,o .jT X =
VE T'Y(o)M· We have the following by solving the initial-value problem for
(7.122).LEMMA 7.83. Given initial data Yo, Y1 E T'Y(o)M, there exists a unique
solution Y (r) of (7.121) with Y (0) =Yo and (\l zY) (0) = Y1.Since (7.121) is linear, the space of .C-Jacobi fields along an £-geodesic
'""(is a finite-dimensional vector space, isomorphic to T'Y(o)M x T'Y(o)M·
REMARK 7.84. If the solution (Mn,g (r) =go) is Ricci flat, then the
.C-Jacobi equation (7.121) says
1
\lx (\lxY) = R(X, Y)X - 2 r \lxY.That is, we obtain the Riemannian Jacobi equation for go,Dv-rx (\l v-rxY) = R (vrX, Y) y!TX;
i.e.,
\l z (\l z Y) = R ( Z, Y) Z.
On the other hand, if g ( r) is Einstein and satisfies Re = .} 7 g, then
3
\lx (\lxY) = R(X, Y) X -
2