- INTEGRAL CURVES TO GRADIENTS OF CONCAVE FUNCTIONS 443
Note that by (H.32), the right tangent vector, if it exists, is unique.
We have the following characterization of the right tangent vector at a
'Lipschitz point' of a path.LEMMA H.41 (Right tangent vector of a continuous path). Let
"(: [a,b]--+ (Mn,g)
be a continuous path and let
"!s,a : [O, O"] --+ M
be a constant speed minimal geodesic joining "( ( s) to "! ( s + O"). If"! is locally
Lipschitz at s E [a, b) in the sense that
d ( "( ( s + O") ' "( ( s)) = 0 ( O") when O" --+ o+'
then we have.... exp~(~)("! (s + O"))
"!+ (s) = hm 'Ys,a (0) = hm
a-to+ a---+O+ O"
provided either 1+ (s) exists or lima---+O+ 1s,a (0) exists.
PROOF .. (1) If 1+ (s) exists. Suppose that, for some s E [a, b), the right
tangent vector i'+ (s) exists. Then, for every smooth function h: M--+ JR,^11
. ( )(h)- d (h )( )- r h('Y(s+O"))-h('Y(s))
"!+ s - ds+ o "( s - a!fi"+ O" •
Since
(H.33) h ("! (s + O")) - h ("! (s)) = 1s,a (0) (h) O" + o (d ("! (s + O"), "( (s)))
= 1s,a (0) (h) O" + 0 (O"),we conclude
1+ (s) (h) = lim 1s,a (0) (h),
a---+O+
so that 1+ (s) = lima---+O+ i's,a (0).
(2) If lima---+O+ 1s,a (0) exists. Now suppose for s E [a, b) the limit
lima-to+ 1s,a (0) exists. Then by (H.33) we have
lim "Ys,a (0) (h) = lim h ("! (s + O")) - h ("! (s)) = 1+ (s) (h).
a-to+ a---+O+ O".
0REMARK H.42. Note that for O" sufficiently small there exists a uniquesmall vector VE T 1 (s)M such that
exp 1 (s) (V) = "( (s + O"),
i.e., exp~(~)·("! (s + O")) is well defined.
(^11) In particular, h is defined on some convex neighbor hood of 'Y ( s) in M and h. is
differentiable at 'Y ( s).