176 4. PROOF OF THE COMPACTNESS THEOREMwhere we are considering the curves Is both in terms of geodesics from x
and geodesics from a ( s). Note that
a~ a~.
as (s, 0) = 0, as (s, 1) =Ct (s)'
a~ -1ar (s, 1) = -expa(s) x.
The second derivative of f has the following expression.
LEMMA 4.44 (Hessian of the distance squared function). The Hessian of
f is given by(\ly gradf) (y) = -\7yexp;^1 x = Va;arl(l),
where 1 ( r) is the 1 acobi field along the geodesic between x and y parametrized
on r E [O, 1] such that 1 (0) = 0 and 1 (1) =YE TyM·PROOF. Given YE TyM, define Is and~ as above. Let ls (r) ~ ~~ (s, r)
be the Jacobi field along Is; then ls (0) = O and ls (1) = a (s). Using
~~ (s, 1) = -exp-;;_(s) x, we compute
\ly exp;^1 x = - \7 a;as ~~ (s, r) I = - (\7 a;arlo) (1).
r (s,r)=(0,1)DThe following is essentially the Hessian comparison theorem.LEMMA 4.45 (Hessian comparison). If the sectional curvature of(Mn,g)is bounded above by K, then there exists a constant C = C ( K) > 0 such
that for any y E B ( x, 7r / ( 2JK)) not in the cut locus we have
(4.14) (Hess f) (Y, Y) = -g (\ly exp;^1 x, Y) ~ C IYl^2 , Y E TyM.
PROOF. By Lemma 4.44 we need to estimateNote that we can write Y = y..L+c')i (1), where y..L is perpendicular to '°)1 (1)
and c ER Then
1(r)=1..L (r) + cr')i (r),where 1..l (r) is a Jacobi field satisfying 1..l (0) = 0, 1..L (1) = y..L and
1..L (r) l '°)1 (r). Now it is clear that we only need to estimate Ill :f;. Ill lr=l
assuming that Y is orthogonal to '°)1 (1).