312 7. THE REDUCED DISTANCE~~ (q, 7) ~~ (q, 7). If L (q, ·) is not differentiable at 7, then by writing
~~ (q, 7) = y'T ( R ('!' (7), 7) + [X (7)[^2 ) - 2v'T [X (7)[^2 , we mean (this is
our convention below) that there is a smooth function L (x, r) satisfying
L (x, r) 2 L (x, r) for x E U and r E (7 - E, 7 + E), L (q, 7) = L (q,T) and
~~ (q, 7) = y'T ( R ('!' (7), 7) + [X (7)[^2 )-2J¥ [X (7)[^2. Now we have provedLEMMA 7.34 (Time-derivative of L formula). The time-derivative of the
£-distance function is given by(7.57) ~~ (q, 7) = -Vf ( R (q, 7) + [X (7)[^2 ) + 2Vf R (q, 7),
where X (7) = ~:;'. (7), for any minimal £-geodesic r : [O, 7] __, M joining p
to q.In the case where (Mn, g ( r) =go) is Ricci flat, we haver (r) = f3 (2..,fi),
where f3 : [o, 2./¥] -r M is a constant speed Riemannian geodesic with
respect to g 0. Thus ~:;'. = }r/J (2..,fi) and l/J (2..,fi) I = d;~. On the otherhand L (q, r) = d~:}J
2- Hence
aL - d(q,p)2 -Id' -12
ar (q, r) = - 473/2 = -Vf dr (r) '
agreeing with (7.57).5. The second variation formula for C and the Hessian of L
Recall that the second variation of arc length formula of a geodesic r is
(7.58) d~ 2 lu=O L bu)
=lb (i\7-yU[^2 - (\7-yU, 1')^2 - (Rm (U, i') i', U)) ds + (\7uU, i') [g,
where 'Yu : [O, b] -r M is parametrized by arc length s and satisfies 10 = r
and U ~ Ju lu=O 'Yu· This formula is fundamental to Riemannian geometry
for a variety of reasons. For example, on a complete Riemannian mani-
fold, any two points can be joined by a minimal geodesic, in which case
l2
u^2 i u=O L bu)^2 0 for endpoint-preserving variations. The second variation
of arc length can also be used to bound from above the Hessian of the dis-
tance function. In this section we consider the analogous second variation
formula for £-length.
The first variation of £-length determines the £-geodesic equation and
is related to the space-time connection. The second variation formula for£-
length at an £-geodesic r: [O, 7] -r Mis related to the space-time curvature
and hence Hamilton's matrix Harnack quadratic. In the case of a minimal£-
geodesic, it also gives an upper bound for the Hessian of the barrier function