314 B. SOME RESULTS IN COMPARISON GEOMETRY
the sectional curvature hypothesis implies exp Pi : B ( 0, 7r / VK) ____, Mn is an
immersion.
Now by the claims above, we know a 00 is a nondegenerate proper 1-gon
or 2-gon, hence belongs to A. It follows that A is a closed subset of the
compact set 3. D
This compactness property implies there is /3 E A whose total length
realizes L (/3) = infrEA L (r). The extra hypothesis of positive sectional
curvature guarantees that /3 is smooth:
LEMMA B.69. Let (Mn,g) be a complete noncompact Riemannian man-
ifold of positive sectional curvatures bounded above by K > 0. Then any
f3 E A such that L (/3) = infrEA L (r) is a smooth geodesic loop.PROOF. By choosing a vertex at L (/3) /2 if necessary, we may suppose
without loss of generality that /3 is a nondegenerate proper 2-gon corre-
sponding to the data (p, V, .e, p', V', £'). Let 'Y denote the path from p to p^1 ,
and let 'Y' denote the path from p' to p. We will show that /3 is smooth at
p', hence at p by relabeling.
Define a 1-parameter family
{f3t: 0 :St< £0 ~min {.e,.e'}}
of nondegenerate proper 2-gons f3t by taking p~ ~ 'Y ( .e - t) and letting 'Yt
denote the truncated path 'Yl[o,e-t]· Then by Corollary B.22, there is a
unique geodesic 'Y~ near 'Y' joining p~ to p. By Proposition B.62, Cs is
totally convex; since p, p~ E Cs, it follows that 'Y~ lies in Cs, hence that
f3t E A for all t E [O, £0). Notice that /3 can fail to be smooth at p' only if
'Y (£) #-'Y' (£'). But the variation vector field of f3t is a Jacobi field V along
'Y' with V (0) = -'Y (£) and V (£') = 0. Thus by the first variation formula,
we take a one-sided derivative to obtain:t L (f3t) lt=O = \V (0), "'f' (0) + /y (£)) = - \ /y (£), "'f' (0)) _ ('Y (£), /y (£))
= - \'Y (£) ,/y^1 (0)) - 1 < 0,
because l'YI = b' I = 1. This contradicts the minimality of L (/3) in A unless
/3 is smooth at p'. DPROOF OF THEOREM B.65. We have already shown that the result can
fail only if there is a smooth geodesic loop /3 of length less than 7r / VK
contained in a compact totally geodesic set Cs based at 0 E Mn. Let a
be any ray emanating from 0. Fort E (1, oo) to be chosen later, join the
loop /3 to the point a ( t) by a minimal geodesic 'Yt : [O, .et] ____, Mn, where
'Yt (0) E /3, 'Yt (£t) = a (t), and .et ~ d (a (t), /3). Since /3 is smooth, we can
apply the first variation formula to conclude that "-ft l_ iJ at 'Yt (0) E /]. As
in the first proof of Proposition B.54, we shall obtain a contradiction by
applying a second variation argument to 'Yt among curves joining /3 to a (t).