310 7. THE REDUCED DISTANCE
4.2. Gradient of L. We compute the gradient of L via the first vari-
ation formula for £. Since L ( ·, T) is not smooth in general, the gradient is
defined in the barrier sense as described below. Let "( : [O, r] ~ M be a
minimal £-geodesic from p to q so that L (q, r) = £ ("!). For any point x in
a small neighborhood U of q and any TE (r - c:, r + c:) with small c: > 0, let
'Yx, 7 : [O, r] ~ M be a smooth family of paths with "fx,T (0) = p, 'Yx,T (r) = x
and 'Yq,'f = 'Y· (Recall that our definition of a smooth variation says that
'Yx, 7 ( u;) is a smooth function of (<7, x, r) .) Define L: Ux (r - c:, r + c:) ~ ~
byL(x,r) =L("fx,T)·
Then L(x,r) is a smooth function of (x,r) when T > 0, L(x,r)::::; L(x,r)
for all (x, r) E U x (r - c:,T + c:), and L (q, r) = L (q, r). That is, the
function L (-, ·) is an upper barrier for L (·, ·) at the point (q, r).
Given a vector Y (r) at q, let q (s) be a smooth path in U with q (0) = q
and ~; (0) = Y (r). Consider the smooth 1-parameter family of paths 'Ys ~'Yq(s),'f: [O,r] ~ M. Let Y(r) ~ Zsls=o'Ys(r) denote the variation vector
field along 'Y (r). By (7.29), (7.32), and Y (0) = 0, we have
\JL A (q,T) · Y (r) = -d di L A (q (s), r) = (Jy£) ("!) = 2vrY '= (r). X (r).
(^8) s=O
Hence
\J. ..t (q, r) = 2-/Tx (r).
It follows from Lemma 7.30 that L (·, r) is differentiable a.e. on M.
Suppose L (·, r) is differentiable at q. Since L (·, r) is an upper barrier for
L (·, r) at the point q, it is easy to see that\7 L (q, r) = \7 L (q, r) = 2-/Tx (r).
Suppose there is another minimal £-geodesic 'Y' : [O, r] joining p to q. Then
we can construct another barrier function L' as above; the same proof will
imply \7 L (q, r) = 2-/TX' (r), where X' (r) = ~~ (r). Now both 'Y and 'Y'
satisfy the same £-geodesic equation and 'Y (r) = 'Y' (r) = q and ~ (r) =
~ (r) = \7 L (q, r). By the standard ODE uniqueness theorem, we conclude
that 'Y (r) = "(^1 (r) for TE [O, r]. Hence if L (·,T) is differentiable at q, then
the minimal £-geodesic joining (p, 0) to (q, r) is unique.
Convention: If the function L (·, r) is not differentiable at q, then
by writing \7 L (q, r) = 2-/TX (r),^8 we mean that there is a smooth
function L satisfying L (x, r) 2: L (x, r) for x E U, L (q, r) =
L (q, r), and \7 L (q, r) = 2-/TX (r).
We have proved the following.(^8) Note that X (f) depends on the choice of minimal .C-geodesic, which may not be
unique.