480 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTIONOur interest in Lipschitz hypersurfaces stems from the following. ·For
the definition of topological end, see p. 143 of Benedetti and Petronio
[11].
PROPOSITION I.41 (Large radii distance spheres are Lipschitz hypersur-
faces). Let (Mn, g) be a complete noncompact manifold with nonnegative
sectional curvature. For s sufficiently large, we have the following.
(i) The distance sphere S (p, s) is a Lipschitz hypersurface.
(ii) The length space ( S (p, s) , .Cds(p,s)) is complete.
(iii) The number of components of S (p, s) is equal to the number oftopological ends of ( M, g). In particular, if ( M, g) has exactly one
topological end, then S (p, s) is connected for s sufficiently large.
PROOF. First we shall prove that if x is a regular point of r, then thereis an open neighborhood W of x such that S (p, r (x)) n W is a Lipschitz
hypersurface.
By Lemma I.33, if x is a (~-)regular point of r, then there exists an
open neighborhood U of x, a smooth unit vector field V defined on U, andan c: > 0 such that for every y E U we have
7r
(I.38) L (V (y), W) :::; 2 - c:for all WE (\7 *r) (y).
Let 'Y be an integral curve of Vin U with 'Y (0) = y for some pointy EU.
By the proof of Exercise I.35, we may deduce
r ('Y (s)) - r (y) 2:: sine:· sfor alls E (0, 6 (y)] and some 6 (y) > 0. In fact,
(I.39) r ('Y (t2)) - r ('Y (ti)) 2:: sine:· (t2 - ti)for any ti < t2 in the domain of 'Y.11
Now let tin-l CU be a smooth hypersurface which passes through x and
which is transversal to V,^12 i.e., for every y E ti we have Ty ti+ ~V (y) =
TyM. Near x, by using V, we shall write S (p, r (x)) as a 'graph' over ti.
It follows from (I.39) that there exists an open neighborhood V c U of
x such that for every point y E ti n V,^13 the maximal integral curve
'Yy: (ay, by) ---t Vto Vin V, with 'Yy (0) = y, also intersects r-^1 (r (x)) = S (p, r (x)) at exactly
one point; call this point <P (y). That is, we have defined a map<P: tin V ---t S (p, r (x)) n V,
where the image <P (tin V) is an open subset of S (p, r (x)).
(^11) We leave it as an exercise to verify (I.39).
(^12) For example, we can take 1-ln-l to be the image of a small (n - 1)-ball in Vi of
exp,,.
(^13) Note that 1-l n Vis nonempty since x E 1-l n V.