1547671870-The_Ricci_Flow__Chow

294 B. SOME RESULTS IN COMPARISON GEOMETRY

1.5. The Rauch comparison theorem. If one has suitable bounds
on the sectional curvatures of a Riemannian manifold (Mn, g), one can
compare distances in Mn with a those in a convenient model space.

THEOREM B.20 (Rauch). Let (Mn,g) and (Mn,g) be complete Rie-

mannian manifolds. Let

r : [O, £] ---+ Mn

i : [O, £] ---+ Mn

be unit-speed geodesics of the same length such that

sect (II) ::; sect (fi)

for any 2-plane II containing T ~ 't and 2-plane fi containing f' ~ ~- Let J
and J be Jacobi fields along r and i respectively, such that J (0) and J (0)
are tangent to r and i respectively, and
[J (0)[ 9 = [J (0)[ 9.
Suppose that

and

(\lrJ (0), T) 9 = ('9 yJ (0), f') 9.

If i (t) is not conjugate to i (0) for any t E [O, £], then for all t E [O, £],

[J (t)[g 2 [J (t) [9.

Note that it is often convenient to apply the theorem when one knows

that sup(sect(Mn,g)) ::; inf (sect (Mn,g)) and takes Jacobi fields J, J

such that J (0) = J (0) = 0, [\lrJ (0)[ 9 = [VrJ (0)[ 9 , and (J, T) 9 =

(J, f') 9 = 0. Note too that the condition (J, T) 9 = 0 implies in particu-

lar that

d

0 = dt (J, T) 9 = (\lrJ, T) 9 ,

because r is a geodesic.

By taking (Al!, g) to be the Euclidean sphere ( S~VK' gcan) of radius

1/VK, one obtains the following result.

COROLLARY B .21. Let K > 0, and suppose (Mn,g) is a complete Rie-

mannian manifold with sect (g) ::; K. If J is a Jacobi field along a geodesic
r in Mn with unit tangent T such that

J (0) = 0 and (\lrJ (0), T) = 0,

then

IJ (t) I 2 [\l~O)I sin ( VKt) > 0

for all t E [O, 7r / VK). In particular, there does not exist a conjugate point

in B (p, 7r /VK) = { q E Mn: d (p, q) < 7r /VK}.