8. .C-JACOBI FIELDS AND THE £-EXPONENTIAL MAP 349we haveD_g,_ (Rc(Y) + \i'yX - I_y)
dr 2T= (! Re) (Y) + 2 (V' x Re) (Y) - 2 Re^2 (Y)
1+ 2 \i'y (V' R) - 2 (V'y Re) (X) - Re (\i'y X)
+ R (X, Y) x + (\i'y Re) (X) - (V' Re) (r, 1")
1 1
+-Y--\i'yX.
2T^2 T
This may be rewritten asD_g,_ (Rc(Y) + \i'yX-I_y)
dr 2T= (:T Re) (Y) + ~\i'y (V'R)-Rc^2 (Y) + ;T Rc(Y)
- 2 (V' y Re) ( X) + 2 (V' x Re) (Y) + R ( X, Y) X
- (Y' Re) ( J{, f) -(V' Re) ( 1;, f)
- Re (Re (Y) + \i'yX - I_y) -~ (Re (Y) + \i'yX - I_y).
2T T 2T
Define the matrix Harnack expressionJ (Y) ~ - (:T Re) (Y) - ~\i'y (V'R) + Rc^2 (Y) -
2~ Re (Y)+ 2 (V'y Re) (X) - 2 (Y' x Re) (Y) - R (X, Y) X- (V'Rc) (1,f) + (V'Rc) (1;,f),
so that (note (-(V'Rc) (J(,f) + (V'Rc) (1;,f)) (Y) = O)
(J (Y), Y) = ~H (·X, Y).
Thus we have the following.
LEMMA 7.85. The £-Jacobi equation is equivalent to(7.124) (Dg, ~ +Re+~) T (Re (Y) + Y'xY - I_y) ~. = -J (Y),
where we have replaced V' y X by V' x Y.EXERCISE 7.86. Rewrite the above equation using Uhlenbeck's trick.