296 B. SOME RESULTS IN COMPARISON GEOMETRY
DEFINITION B.24. If 1 : [0, oo) ----+ Mn is a geodesic with 1 (0) = p, we
call 1 ( t 1 ) the cut point of p along 1 ·
Notice that if Mn is compact, there is for each p E Mn a unique cut
point of p along every geodesic 1 emanating from p.
DEFINITION B.25. The union
Cut (p) ~ b (t 1 ): 1: [O, oo)----+ Mn is a geodesic with 1 (0) = p}of all cut points along all geodesics emanating from p E Mn is called the
cut locus of p.
The existence of a conjugate point is sufficient (but not necessary) for a
geodesic to fail to minimize distance.
LEMMA B.26. Let 1 : [O, £] ----+ Mn be a geodesic. If there is T E (0, £)
such that 1 ( T) is conjugate to 1 ( 0) along 1, then there exists a proper vari-
ation
{f s ( t) : -E < S < c, 0 '.S t '.S £}
with I's (0) = 1 (0), I's(£)= 1 (£), and I'o = 1 such that for some s I-0,
L (I's)< L (1).
Briefly: a geodesic fails to minimize distance past its first conjugate point.
COROLLARY B.27. If 1 (t) is conjugate to 1 (0) along 1, then t ;:::: t 1.
If expP does not fail to be an immersion at the cut locus, it must at least
fail to be an embedding there. This statement can be made more precise.
LEMMA B.28. 1 (t) is a cut point of 1 (0) along the unit-speed geodesic1 if and only if t > 0 is the smallest value such that either of the following
(non-exclusive) conditions hold:
(1) The point 1 (t) is conjugate to 1 (0) along --y.(2) There exists a unit-speed geodesic (3 : [O, t] ----+ Mn such that (3 (0) =
1 (0), /3 (t) = 1 (t), and L ((3) = L ('Yl[o,tJ), but /3 (0) I-~ (0). That
is, (3 and 11 [O,t] are distinct geodesic paths of equal length from their
common starting point 1 (0) to their common end point 1 (t).
Given p E Mn, denote the unit sphere in TpMn by
s;-^1 Mn ~{VE TpMn: IVI = 1}'
and define p: s;-^1 Mn----+ (0, oo] by(V) ~ { d (p, 1V ( tw)) = tw if t 1 v e_xists
p oo otherwise.
That is, p (V) is the distance from p to the cut point of p along 1v.
LEMMA B.29. For each p E Mn,
p : s;-^1 Mn ----+ (0, oo]
is a continuous function. That is to say: