- CRITICAL POINT THEORY AND PROPERTIES OF DISTANCE SPHERES 475
PROOF. (1) If case. Suppose that VE s;i-^1 , c > 0, and c >Oare such
that
r (expx (sV)) - r (x) ?: cs
for all s E [O, c). Let bsLE[O,c) be any smooth family of paths joining p to
expx ( s V) with 'YO being a minimal geodesic. ThenLbs) ?: r (expx (sV)),
while L ('Yo)= r (x). On the other hand, by the first variation of arc length
formula, we have(v, 1' .)--5!_1 - d L( 'Ys )>l' Imm .fr(expx(sV))-r(x)> _ c >. 0
s s=O s-to+ s
Since 'YO is an arbitrary minimal geodesic joining p to x, we conclude that x
is a regular point of r.(2) Only if case. Suppose that xis a regular point of r. By definition,
there exists VE s;i-^1 such that for every WE (V'*r) (x) we have
1fL(V, W) <
2
.
Let a (s) ~ expx (sV) for s E [O, c). By Lemma I.33 and a (0) = V, provided
c > 0 is sufficiently small, we have
(I.26) L (a ( s) , W) ~ i -c
for all WE (V'*r) (a (s)) ands E [O,c).
Since r is Lipschitz, we have r o a : [O, c) -+ [O, oo) is Lipschitz and hence
(r o a)' (s) exists for a.e. sand for any s E (0, c) we have^8(r o a) (s) - r (x) =las (r o a)' (s) ds.
Given s 0 E (0, c) such that (r o a)' (so) exists, we have
( )
'( )
1
. (roa)(so)-(roa)(so-.6.s)
r o a so = im A •
As-to+ us
For o > 0 sufficiently small, let {,8As} AsE[0,<5) be any smooth family of
paths joining p to a (so - .6.s) with ,80 being a minimal geodesic. Clearly
L (,8As) ?: (r o a) (so - .6.s ). Hence
(r o a)' (so)?: lim L (,Bo)~ L (,8As) = / ~o (a (so)), a (so)),
As-to+ s \where the last equality follows from the first variation of arc length formula.
Since ~o (a (so)) E (V' *r) (a (so)) and by (I.26), we conclude that
(r o a)' (so)?: cos (i -c) > 0
(^8) Note that, a priori, r may not be differentiable at a (s) for a.e. s.