180 4. PROOF OF THE COMPACTNESS THEOREM
Then (fk(l,y,z))
theorem to
(xk) = expy z. We will apply the implicit function
F (x, y, z) ~ f (1, y, z) - x
to prove that z = expy-^1 x is a smooth function of (x, y) and that exp;^1 x
has the required derivative estimates. Consider the boundary value problem
for the first-order ODE
dfk = hk
dr 'dhk + r~. (!) hihj = o
dr iJ 'fk (0,y,z) = yk,
hk (0, y, z) = zk.
From J(r,y,z) E B(p,r1), Jh(r,y,z)J 9 = JzJ 9 (p) = d(y,x), and ~(Iij)::::::;
(gij), we have IJk(r,y,z)I::::::; V2r1 and [hk(r,y,z)I::::::; V2d(y,x). From
the smooth dependence property for ODE (see Theorem 4.1 on p. 100 of
Hartman [196]), Jk (r, y, z) is a smooth function of (r, y, z). Using the proof
of Theorem 3.1 on p. 95 of [196] and (4.19), it follows from an induction
argument on the order of derivatives that for r E [O, 1] ,
(4.21)()CY.+/3
(8y)°' (8z)/3fk (r, y, z) ::::::; C[af+l/31+1'aa+/3
(8y)°' (8z)/3hk (r, y, z) ::::::; Claf+l/31+1·Actually the proof of Theorem 3.1 on p. 95 of [196] implies the estimate
above for Jal+ J,8J = 1.
Let ry,z be the geodesic with /y,z (0) = y and (d~ ry,z) (0) = z. Let
ly,z,'i (r) denote the Jacobi field along the geodesic ry,z with ly,z,'i (0) = 0
and (j;.Jy,z,z) (0) = z E TyM. Then the covariant partial derivative in the
direction z is
Dzf (1, y, z) (z) = !f (1, y, z + sz) ls=O = ly,z,z (1).
To show Dzf (1, y, z) : TyM----+ Texpy zM is invertible, we prove that there is
a constant co> 0 such that Jly,z,'i (l)J 2: co JiJ. This follows from the Rauch
comparison theorem; here we give a proof using (4.15). As in the proof of
Lemma 4.45, it suffices to prove that J ly,z,'i (1) J 2: co JiJ for those z which
are orthogonal to z in TyM· From (4.15), we have (!Jy;(:r)I)' 2: 0, where
'!-' ,/, ( ) r = 1-1 z snColzl2 ( ) r. s· mce l" Imr-->O !Jy,z,.z(r)J
Jly,z,zl (1) 2: </> (1) = JzJ snColzl2 (1) 2: co JzJ.