- CRITICAL POINT THEORY AND PROPERTIES OF DISTANCE SPHERES 477
for all i E N. Then for all i there exist Vi, Wi E (\7 * r) (Xi) such that
(I.29) L (Vi, Wi) > 2c.Note that the unit speed geodesics
defined by
'°YVi : [O, ri] ---7 M,
/1Wi : [O, ri] ---7 M,
'°YVi (s) ~ expxi (-h - s) Vi),
/1Wi (s) ~ expxi (-(ri - s) Wi)
for s E [O, ri], both join p to Xi.
Fix an E > 0 and an i EN. Suppose that x EM is such that
r·
(I.30) r (x) = d (x,p) 2: _i •
1-E
Let di ~ d (x, Xi) and let
CTi: [O,di] ---7 Mbe a unit speed minimal geodesic joining Xi to x. Note that from the triangle
inequality and (I.30), we have
E
(I.31) di 2'.r(x)-d(xi,P) 2:
1_·Eri.By (I.29), we may assume, without loss of generality (i.e., by switching Vi
and Wi if necessary), that
(I.32)where c E (o, ~) is as above.
Now define
By (I.32),
(I.33)
Let (3: [O, r (x)] ---7 M be a unit speed minimal geodesic from p to x and
let
</>i ~ ,{_ (~ (0) ''YVi (0)).We have a geodesic triangle b..pxix in M with sides ai, (3, and '°YVi and
corresponding opposite vertices p, Xi, and x, respectively. Consider the
comparison triangle LS. in Euclidean space with side lengths r (x), ri, and di
which is associated to the geodesic triangle b..pxix in M. We denote the
angles of LS. corresponding to </>i and (Ji by ef>i and ei.
By the triangle version of the Toponogov comparison theorem (Theorem
G.33(1)), we have(I.34) ;;.,. 'f'i <k - <pi