- FIRST VARIATION OF £-LENGTH AND EXISTENCE OF £-GEODESICS 297
LEMMA 7.15 (.C-First Variation Formula). Given a 0^2 -family of curves'Ys : [r1, r2] -+ N, the first variation of its .C-length is given by
~ (oy.C) ('Ys) ~ ~ :s.c bs) = VTY ·XI~~
(7.29) + VTY · -\JR--X - \lxX - 2Rc (X) dr,
1T2 (1 1 )
Tl 2 2Twhere the covariant derivative \1 is with respect to h ( T).REMARK 7.16. We use the notation (oy.C) ('Ys) since Js.C ('Ys), at a given
value of s, depends only on 'Ys and Y along 'Ys·PROOF. We compute in a similar fashion to the usual first variation
formula for length (see [72], p. 4ff for example)(7.30) d d 1T2 ( ·^1 a^1
2
-d .C ('Ys) = -d VT R ('Ys (r), r) + 28 (r) ) dr
S S Tl UT h(T)1
T2
= VT ( (\1 R, Y) + 2 (\1 y X, X)) dr ;
Tlhere ( · , · ) = h ( T) ( · , · ) denotes the inner product with respect to h ( T).
Using [X ' Y] = [^0 OT G ' oG] as = 0 and 2-h OT = 2Rc ' we haved
(\ly X, X) = (\1 x Y, X) = dr [h (Y, X)] - (Y, \1 xX) - 2 Re (Y, X).Hence~~.c ('Ys) =1T
2
VT(~ (\JR, Y) + dd (Y, X) - (Y, \lxX) - 2Rc (Y, x))dr
2~ n 2 Tand integration by parts yields1
T2 d 11T2 1
VT-d (Y, X) dr = -- r,:;. (Y, X) dr + VT (Y, X) [~~.
Tl T 2 Tl VT
The lemma follows from the above two equalities. DREMARK 7.17. In comparison, the Riemannian first variation of arc
length formula on (J.An, g) is(7.31)d rb b
du L ('Yu)= -Jo (U, \!TT) ds + (U, T)[ 0 ,where 'Yu : [O, b] -+ M is a 1-parameter family of paths, T ~^0 {Lu j I^0 {Lu I '
U ~ tu 'Yu, and ds is the arc length element.