290 B. SOME RESULTS IN COMPARISON GEOMETRY
is singular. In this case, we say q is conjugate top along the geodesic IV
defined by IV : t f------7 expP ( t V).
DEFINITION B .4. The conjugate radius re E (0, oo] of a point p E Mn
is
re (p) ~ sup { r : ( expp) * is nonsingular in B(O, r)}.
REMARK B.5. expPIB(O,rc) is an immersion of the open ball B(O, re)·Intuitively, a conjugate point is one where distinct geodesics come to-
gether. The following result (which uses terminology from Definition B .66)
makes this notion precise.
LEMMA B.6. Let (Mn, g) be a complete Riemannian manifold, and let
{ri}iEN be a sequence of nondegenerate proper geodesic 2-gons,ri = { 1f : [o, £i] ---t Mn, I~ : [o, £~] ___,Mn},
such that 1i (0) = 11~ (£~) ~Pi and I~ (0) = 1i (£1) ~ qi. Suppose that the
following limits all exist:
p 00 ~ .lim Pi E Mn,£ 00 ~ i--+ lim (X) £i = i--+ lim (X) £~.
Then £ 00 > 0, and q 00 is a conjugate point to p 00 along the limit geodesic
'Yoo: [0,£ 00 ] ----t Mn determined by 1oo (0) = V 00.
PROOF. That £ 00 > 0 follows from the lower bound for the injectivity
radius of (Mn, g) in a compact set. Now suppose that q 00 = expp 00 (£ 00 V 00 )
is not a conjugate point of p 00 along the geodesic loo : [O, £ 00 ] ---t Mn. Then(expPooL: TR.ooVoo (TpooMn) ---t Tq 00 Mn
is nonsingular. Since exp : T Mn ---t Mn is a smooth map, there exists a
neighborhood U of p 00 in Mn and a neighborhood V of £ 00 V 00 in Tp 00 Mn
such that
(expPL: Tw (TpMn) ---t Texpp(w)Mnis nonsingular for all p E U and W E V c TpMn. Here we used parallel
translation along geodesics emanating from p 00 to identify Tp 00 Mn with
TpMn for all p EU, which allows us to regard Vas a subset of TpMn. Thus
it follows from the inverse function theorem that there exists a neighborhood
V' ~ V and some E > 0 such that
expPIB(R.oo Voo,c:) : B (foo Voo, E) ---t Mn