- GRADIENT AND TIME-DERIVATIVE OF THE £-DISTANCE FUNCTION 311
LEMMA 7.32 (Gradient of L formula). The spatial gradient of the L-
distance function is given by(7.54) v L (q, r) = 2v1f x (r),
where X (r) = ~;'. (r), for any minimal £-geodesic 'Y : [O, r] ---+ M joining
p to q. Furthermore if L (·, r) is differentiable at q, the minimal £-geodesic
joining (p, 0) to (q, r) is unique.REMARK 7.33. The analogy of (7.54) in Riemannian geometry is asfollows. Let dp (x) ~ d (x,p) and suppose dp is smooth at q E (J\/tn, g).
Define 'Y : [O, b] ---+ M to be the unique unit speed minimal geodesic from p
to q. By the first variation formula (7.31), for any U E TqM,
(\ldp (q), U) = d~ 'u=O L ('Yu)= ('Y (b), U),
provided the 'Yu: [O,b]---+ M satisfy 'YO="(, 'Yu (0) = p and a°ulu=o'Yu (b) =
U. That is, \ldp (q) = i (b).
Taking the norm of (7.54),(7.55) IVLl^2 (q, r) = 4r IX (r)l^2 = -4rR (q, r) + 4r ( R (q, r) +IX (r)l^2 ).
The reason we rewrite this in a seemingly more complicated way is that both
Rand R +IX (r)l^2 are natural quantities.^94.3. Time-derivative of L. Next we compute the time-derivative of
L. This time we need to choose 'Yx,T used in subsection 4.2 above a little
more carefully. Given (q, r), let 'Y : [O, r] ---+ M be a minimal £-geodesic
from p to q so that L (q, r) = £ (1). We first extend 'Y to a smooth curve
'Y: [O, r + c-] ---+ M for some c > 0, and then we choose a smooth family of
curves 'Yx,T to satisfy 'Yx,T (0) = p, 'Y"f(T), 7 = 'Yl[o, 7 ] and 'Yx,T (T) = x. Define
(7.56) L (x, T) ~ £ bx,T)
for ( x, T) E U x ( r - c-, r + c-).
We compute, using the chain rule and (7.54),
= d~'T=T [L('Y(T),T)]-vt.x
T=ToL ( _) _ oL ('Y (T), T)
OT q,T - OT~ : 7 I,~, [f v'f ( R ('Y (f), r) + j ~; (r)j}r] -2v'f 1x (f)I'
= vlf ( R ('Y (r), r) +IX (r)l^2 ) - 2vlf IX (r)l^2.
It follows from Lemma 7.30 that L (q, ·) is differentiable a.e. on (0, T).
As discussed in subsection 4.2 above, if L (q, ·) is differentiable at r, then
(^9) As it is the integrand of the £-length, ../i (R + IXl^2 ) is a natural quantity.