- THE SECOND VARIATION FORMULA FOR L AND THE HESSIAN OF L 315
where we used the £-geodesic equation (7.32) to get the last equality; in the
above,V'f,yR ~ Y (Y (R)) - (\i'yY) (R) =Hess (R) (Y, Y).That is,LEMMA 7.35 (£-Second variation - version 1). Let 7 E (0, T) and let"'( : [O, 7] ---+ M be an £-geodesic from p to q and let Y ~ gs "'is for some
smooth variation "'is of"'( with Y (0) = 0. The second variation of £-length
is given by(5f.L:) ("'!) = 2Vf (\i'yY,X) (7)
( 7. 62 ) + {7 y'7 ( V'f,yR + 2 (R (Y, X) Y, X) + 2 f\i'y Xf
2
) dT.
Jo -4 (\i'y Re) (Y, X) + 2 (V' x Re) (Y, Y)REMARK 7.36. Note that by (7.54) we have (5£) ("'!) = 2yf;jX (7). Hence(5f.C) ("'!) - 2Vf (X, \i'yY) (7) = (5f.C) ("'!) - (5'VyYC) ("'!),
whose value only depends on Y ( T) defined along "'! ( T). This is analogous to
considering (Hess f) (Y, Y) = YY (!) - \i'y Y · \i' f.We now rewrite the £-second variation formula in a better form, which
relates to Hamilton's matrix Harnack quadratic, i.e., the space-time curva-
ture.11 Since
d~ [Re (Y ( T) , Y ( T))] = ( :
7Re) (Y, Y) + (V' x Re) (Y, Y) + 2 Re (V' x Y, Y) ,
integrating by parts, we have- for VT (:T Re) (Y, Y) dT
= for y'7 (
2~ Re (Y, Y) + (V' x Re) (Y, Y) + 2 Re (V' x Y, Y)) dT
- y'TRc (Y, Y)I~.
llSee Section 5 of Chapter 8 for the reason why the space-time curvature is Hamilton's
matrix quadratic.