1547671870-The_Ricci_Flow__Chow

298 B. SOME RESULTS IN COMPARISON GEOMETRY

further subsequence, we may thus assume Vi' -t V~ #-V. By continuity of

the ~xponential map, this implies that expP (p 00 V~) = expP (Poo V) and

d (p, expP (p 00 V~)) = .lim d (p, expP (p (Vi) Vi'))

i--->oo

= .lim d (p, expp (p (Vi) Vi)) = d (p, expp (p 00 V)) ,

i--->oo
which contradicts the fact that expp (p 00 V) is not a cut point of p along the
geodesic expP ( t V). D

COROLLARY B.30. Cut (p) is a closed set for each p E Mn.

Given p E Mn, define

Gp~ {VE TpMn: d (p, expP (V)) = IVI}.

Recalling that /V (t) = expP (tV), we observe that

Gp= {tV: VE s;-^1 Mn and d(p,/v (t)) = t}

= {tV: VE s;-^1 Mn and t ~ d(p,/v (t 1 'V))},

hence conclude that Gp is closed.

DEFINITION B.31. If p E Mn, we call 8Gp C TpMn the cut locus in

the tangent space TpMn.

Note that 8Gp may be the empty set 0. But in any case,

Cut (p) = expp (8Gp).

Moreover, Mn\ Cut (p) is homeomorphic to Gp\8Gp.

LEMMA B.32. For each p E Mn, the map

expPlintCp : Gp\8Gp -t Mn\ Cut (p)

is an embedding.

Note that if Mn is compact, then int Gp, hence Mn\ Cut (p), is homeo-

morphic to an open n-ball.

DEFINITION B.33. The injectivity radius inj (p) at a point p E Mn is

inj (p) ~sup {r > 0: expPIB(o,r) : B (o, r) -t Mn is an embedding}.

The injectivity radius inj (Mn, g) is

inj (Mn,g) ~inf {inj (p): p E Mn}.

When we want to take limits of a sequence of Riemannian manifolds, it

is of fundamental importance to be able to estimate the injectivity radius

from below. A basic estimate is the following.

LEMMA B.34 (Klingenberg). If Mn is compact and sect (g) ~ K for

some constant K > 0, then

in· (Mn ) > min { ___!!_ ~ ( the length of the s~ortest ) }.

J 'g - .jK' 2 smooth closed geodesic in Mn