1547671870-The_Ricci_Flow__Chow

1. SOME RESULTS IN LOCAL GEOMETRY 291

is injective for all p E V'. Now observe that for all sufficiently large i, we
have

Pi EV',

£h1 (0) E B (C=V 00 ,c) c Tp;Mn,

• Ch~ (C~) EB (C= V=, c) c Tp;Mn.

Since Cbf (0) -/= -Ch~ (CD but expPi [Cb{ (0) J = expPi [-Ch~ ( C~)] = %

this contradicts the injectivity of expPIB(t'oo Voo,c:)° D

REMARK B. 7. It is important to note that this result can fail in the con-

text of sequences (Mi,gi) of Riemannian manifolds - in particular, when

each ri is a nondegenerate proper geodesic 2-gon in (Mi, 9i). The failure

can occur even if the sequence (Mi, 9i) has uniformly bounded geometry.

This fact becomes highly relevant when one wants to develop injectivity

radius estimates for such sequences in order to pass to a limit (M~, 9=)·

EXAMPLE B.8. Consider a sequence { T;,^2 } i = Z+ of collapsing fl.at tori with

fundamental domains

[-i, i] x [-1/i, 1/i] c n~.2.

Take Oi = (0 , 0), and define constant-speed geodesics

O'.i, /3i : [ 0, Ji2=1/i] --+ T;,^2

by

ai ( s) = ( s, csc-^1 ( i) · s) and /3i (s) = (s, - csc-^1 (i) · s).

Then length ai = length /3i = 1 for all i, but their limit in the universal cover

JR^2 is just the segment s i---+ (s, 0) for 0 :::::; s :::::; 1, which has no conjugate

points.

EXAMPLE B.9. Consider Sf = { (x , y , z) : x^2 + y^2 + z^2 = 1}. For each

i E N, let (MT, 9i) denote Sf/Zi, where Zi acts on Sf by rotation by angle

27r / i around a fixed axis, say the x-axis. The surface Mr is an orbifold with

two cone points of order i. Simply for the sake of visualization, isometrically

embed MT into JR^3 as a surface of revolution about the x-axis centered at

the origin. This embedding will be smooth except at ,the two cone points

which are at opposite ends of the surface and lie on the x-axis. Since Mr

is collapsing, its length along the x-axis tends to 7r. For i large enough,

consider two points in Mr defined as follows. Let p} = ( x}, y[, z[) be the

unique point on Mr with x} = -1, y[ = 0, and z[ < 0. Let PT = ( x7, YT, zf)

be the unique point on MT with x7 = +1, YT = 0, and z[ > 0. There are

exactly two minimal geodesics ai and /3i joining p} and PT in Mr. These

geodesics have the same length, and the angle between them at p} (and pT)

tends to 0 as i--+ oo. However, the limit

'Y= = t ---tHm OO ai = t ---tHm OO /3i