- CRITICAL POINT THEORY AND PROPERTIES OF DISTANCE SPHERES 473
The set V' *r C SM is locally compact. In particular, if M is compact, then
so is \i'*r.PROOF. Let Xi EM denote the basepoints of 1/i, which converge to the
basepoint x 00 of Voo. Let ri = d (xi,p). ThenIi (s) ~ expxi (-(ri - s) 1/i),
where s E [O, ri], is a unit speed minimal geodesic joining p to Xi such that~i (ri) = 1/i. Let
loo (s) ~ expx 00 (-(roo - s) V 00 ),
wheres E [O,r 00 ] and r 00 = d(x 00 ,p). Then Ii--+ loo and so loo is a
unit speed minimal geodesic joining p to x 00 with ~ 00 (r 00 ) = V 00 • Hence
V 00 E V'*r. DHaving defined the gradient of r, the following definition is natural.DEFINITION I.31 (Regular point of a distance function). (i) We say that
a point x E M is a regular point of r ~ d (x,p) if there exists V E s:;;-^1
such that for every W E (V' * r) ( x) we have
7r
(I.24) ,{ (V, W) < 2'i.e., (V, W) > 0.
(ii) We say that x E M is a critical point of r if it is not a regular
point of r; i.e., x is a critical point of r if and only if for every V E s;;-^1
there exists W E (V' * r) ( x) such that
7r
,{ (V, W) 2:
2.
Note that if x EM is a regular point of r, then (V'*r) (x) c s~-·^1 is
contained in an open hemisphere. Actually, since (V' *r) (x) is compact, for
the vector Vin (I.24), there exists c: > 0 such that
7r
(I.25) ,{ (V, W) ::; 2 - c:for all WE (V' *r) (x).
Since V' r is a locally compact set, the set of critical points of r is a
closed set. In other words, the set of regular points of r is an open set.
The following are trivial examples of critical points:
(1) the point p itself is a critical point of r since, by definition,
(V'r) (p) = s;-^1 ,
(2) any farthest point from pis a critical point of r,
(3) more generally, any local maximum of r is a critical point of r,( 4) if there exists W E s~-l such that W, -W E V' *r, then x is a
critical point of r. Indeed, for every VE s:;;-^1 , either,{ (V, W) 2: ~
or,{ (V, -W) 2: ~·