1547671870-The_Ricci_Flow__Chow

292 B. SOME RESULTS IN COMPARISON GEOMETRY

in the limit geodesic tube, which has constant curvature 1, is a geodesic of
length 2. In particular, loo does not have any conjugate points along it.

1.3. Jacobi fields. We have already seen how distinct geodesics can

come together at a conjugate point. The infinitesimal version of this idea is

captured in the concept of a Jacobi field.

DEFINITION B.10. A Jacobi field is a variation vector field of a one-

parameter family of geodesics.

LEMMA B.11. expP (V) is conjugate top if and only if there exists WE

TpMn{O} such that (W,V) = 0 and (exppL(Wv) = 0, where Wv E
Tv (TpMn).

PROOF. Sufficiency is clear. To prove necessity, it is enough to observe

that iv (1) E Texpp(v)Mn\{O}. D

PROPOSITION B.12. Let (Mn,g) be complete and connected. Then q is

conjugate top along a geodesic/ if and only if there exists a nonzero Jacobi

field J along / with

J (0) = J (1) = o.

PROOF. By Hopf- Rinow, q = expp (V) for some VE TpMn\ {O}. Take

any W E TpMn\ {O} such that (W, V) = 0. For s small, consider the

one-parameter family of geodesics

'Ys : t f--+ expp [t (V + sW)].

Note that / = 10 is a geodesic such that / (0) = p and I (1) = q. The

corresponding Jacobi field is given by

Jw (t) ~ ~ 'Ys (t)I

us s=O

= :s expp (tV + stW)ls=O = (expPL (tWtV) E T'Y(t)Mn.

Since Jw (0) = 0 and

Jw (1) = (expPL (Wv),

the result follows from the lemma. 0

By using the calculus of variations, one obtains the Jacobi equation.

LEMMA B.13. J is a Jacobi field along a geodesic 'Y with unit tangent

vector T if and only if

\lr\lrJ = R (T, J) T.

COROLLARY B.14. Any Jacobi field J along a geodesic 'Y with unit tan-

gent vector T admits the unique orthogonal decomposition

J = Jo + (at + b) T,