356 7. THE REDUCED DISTANCE
That is, if Ty < T, then 'TV is the first time the £-geodesic ')'V : [O, T) -+
M stops minimizing. On the other hand, 'TV = T if and only if 'f'V is
minimal. The lemma establishes that either ')'V is minimal or there exists a
first positive time Ty past which '/'V does not minimize.
Let 'Y: [O, T) -+ M be an £-geodesic with 'Y (0) = p. We say that a point
('!' (7), 7), 7 E (0, T), is an £-conjugate point to (p, 0) along ')' if there
exists a nontrivial £-Jacobi field along ')' which vanishes at the endpoints
(p, 0) and (1'(7), 7). A point ( q, 7) is an £-conjugate point to (p, 0) if ( q, 7)
is £-conjugate to (p, 0) along some minimal £-geodesic 'Y ( T) , T E [O, 7], from
p to q. If 'Y ( T) = '/'V ( T) for some V E TpM, then this is equivalent to V
being a 'critical point of the £-exponential map .C 7 exp (see Corollary 7.93).
DEFINITION 7.96. (i) The .C-cut locus of (p, 0) in the tangent space
of space-time is defined by
.CC(p,o) ~ {(V, 'TV) : VE TpM},
where 'TV is defined above. Since TV> 0, we have .CC(p,O) C TpM x (0, T].
(ii) The .C-cut locus of (p, 0) at time 7 E (0, T] in the tangent space
is defined by
(iii) Definen(p,0) (7) ~ {V E TpM : 'TV > 7}.
In words, n(p,O) (7) is the open set of tangent vectors at p for which
the corresponding £-geodesic minimizes past time 7. Note that n(p,O) (7) is
not necessarily star-shaped in the sense that if V E n(p,o) (7), then aV E
D(p,O) (7) for any a E (0, 1). However D(p,O) (7) is an open subset in TpM
and f! ( T2) C f! ( T1) if 'Tl < T2.
Next we define the .C-cut locus of the map .C 7 exp for 7 E (0, T).
DEFINITION 7.97. (i) The .C-cut locus of (p, 0) at time 7 is defined
by
.C Cut(p,O) (7) ~ {.C 7 exp (V) : VE TpM and Ty = 7}.
(ii) We define .C Cut(p,O) (f) to be the set of points q E M such that there
are at least two different minimal £-geodesics on [O, 7] from p to q.
(iii) We define .C Cut(p,O) (7) to be the set of points q such that (q, 7) is
£-conjugate to (p, 0).EXERCISE 7.98. Show that for 7 E (0, T) we have q tj_ .C Cut(p,O) (7) if
and only if for every minimal £-geodesic ')' : [O, 7] -+ M joining p to q, we
may extend 'Y as a minimal £-geodesic past time 7.There is a characterization of .C-cut locus points analogous to the char-
acterization of cut locus points in Riemannian geometry. The first lemma
below can be proved using the locally Lipschitz property of the £-distance
L, while the second lemma below can be proved via a calculation similar to