398 8. APPLICATIONS OF THE REDUCED DISTANCEfor some constant C4 < oo. From (8.34), (8.35), and (8.31), it is easy to see
that J(q, r, h)J is bounded by an integrable function on M, independent of
h small enough. DEXERCISE 8.21. Let (Mn,g) be a Riemannian manifold, p EM, and
assume that expP: TpM --t Mis a diffeomorphism (Mis then diffeomorphic
to ]Rn). Define
rps: M --t M
by
rps : expP (V) 1-t expP ( esV)for VE TpM. Show that {rpsLE!Rt is a I-parameter group of diffeomorphisms
and ·where r (x) ~ d (x,p).SOLUTION TO EXERCISE 8.21. Another way to define rps is
rps (x) = expp (es exp_;^1 x).
We have
rps 1 ( rps 2 (x)) = expp ( es^1 exp_;^1 o expp ( es^2 exp_;^1 x))
= expp ( es^1 +s^2 exp_;^1 x) = rps 1 +s 2 (x) ·Noter (rps (x)) =es iexp_;^1 xi. We compute
( :s rps) (x) = ( d expp) e• expj;" 1 x (es exp_;^1 x)
= (r\7r) (rps (x)) = \7 ( r:) (rp 8 (x)).
Now we consider a Ricci flow analogue of the above discussion. Given p E
M and To > o, assume that n(p,0) (ro) = TpM. For T such that n(p,0) (r) =
TpM, define
</>T: M --7 M
bythat is,
We compute
(! </>T) (x) = ( : 7 CT expP) ((£ 70 expP)-
1
(x))= X (</>T (x)) = (\7£) (</>T (x)),
where X ~ d~ "(v (r) for V = (£ 70 expP)-^1 (x).