1547671870-The_Ricci_Flow__Chow

302 B. SOME RESULTS IN COMPARISON GEOMETRY

is continuous in a neighborhood U ~ Mn\ Cut (p) of p and is smooth on

U\ {p}. Let

"(: [O, inj (p))---> Mn

be a unit-speed geodesic such that 'Y (0) = p and /.y = \Jr, and fix any

r E (0, inj (p)). By (B.3), we obtain the differential inequality

1

< du (r) = (\Jr) (u)

• 1 + Bu^2 1 + Bu^2 '
  which implies
  r < r /.y ( t) ( u) dt


• lo l+B·u('Y(t))^2
  So if B > 0, we get

rd ( 1 ) tan-

1

( VBu('Y(r)))

r :S lo dt VB tan-

1

( v'Bu ('Y (t))) dt = VB ;

while if B = 0, we have

r :S for ~~ dt = u ('Y ( r)) ;

and finally if B < 0, we obtain

1

r d ( 1 _ 1 ( t--,=; ( ( )))) tanh-

1

r:S -d ( FBu ('Y (r)))

o t v - B^115 tanh v -Bu 'Y t dt= v - B^115.

The result now follows easily from the monotonicity of tan - l and tanh-^1. D

2.4. The Toponogov comparison theorem. The Rauch comparison

theorem works at infinitesimal length scales to compare the geometry of a

Riemannian manifold (Mn, g) with model geometries of constant curvature.

It has a powerful analog at global length scales: the Toponogov comparison

theorem.

THEOREM B.40 (Toponogov Comparison Theorem). Let (Mn,g) be a

complete Riemannian manifold with sectional curvatures bounded below by

HER

Triangle version (SSS): Let~ be a geodesic triangle with vertices (p, q, r),

sides qr, rp, pq of lengths

a = length (qr) , b = length ( rp) , c = length (pq)

satisfying a :S b+c, b :S a+c, c :S a + b (for example, when all of the geodesic

sides are minimal), and interior angles a = Lrpq, /3 = Lpqr, 'Y = Lqrp,

where a,/3,"f E [O,n]. Assume that c :S n/VH if H > 0. (No assumption

on c is needed if H :S 0.) If the geodesics qr and rp are minimal, then there

exists a geodesic triangle LS. = (p, ij, r) in the complete simply-connected space