1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

8. £.-JACOBI FIELDS AND THE £-EXPONENTIAL MAP 359

Rearranging terms and integrating by parts, we express this as follows. De-

fine the symmetric 2-tensor Q by

Q (Y, W) ~ ( :T Re) (Y, W) + ~ \7 y \7 w R - (Re (Y) , Re (W))

1.

+ 2 T Re (Y, W) + (R (Y, X) W, X) - (\7y Re) (W, X)

- (\7 w Re) (Y, X) + 2 (\7 x Re) (Y, W)

and

1

S (Y) ~ Re (Y) + \7 x Y -

2

T Y.

Then we have

LEMMA 7.102 (£-index form). The £-index form £I(Y, W) can be writ-

ten as

1

'Tb

£I(Y, W) = - vrRc (Y, W)I~~ + ra yrQ (Y, W) dT

1

rb ( (S (Y) , S (W)) + ( S (Y) , ~) )

+ 0 +1Y \ 2r' s(w)) + (Y,w> 4r2 dT.

Ta

Note that, assuming [X, Y] = 0 and [X, W] = 0, we have

d~ ( (Y,TW)) = ~ (Re+ Sym (\7 X) -

2

~ g) (Y, W) ,

where Sym (V'X)ij ~! (V'iXj + Y'jXi) = !£xg.

8.5. £-Jacobian. First we recall the Jacobian in Riemannian geome-

try. Let (Mn,_q) beaRiemannianmanifold, letp EM, a~dgiven VE TpM

with IVJ = 1, let 'YV : [O, sv) --+ M be the maximal unit speed minimal

geodesic with 1 (0) = V. Take {Ei}r,: 11 to be an orthonormal frame at p per-

pendicular to V and define Jaco bi fields {Ji ( s)} along 'YV so that Ji ( 0) = 0

and (V'vJi) (0) = Ei· The Jacobian J is defined by

J ( 'YV ( S)) ~ J det ((Ji( s) , Jj ( S)) ) ,

where ( (Ji ( s) , Jj ( s))) is an ( n - 1) x ( n - 1) matrix. Note that

(7.137) lim J ('Yv (s)) = 1.

s-+0 sn-1

Let dO"sn-1 denote the volume form on the unit (n - 1)-sphere in TpM,

which naturally extends to TpM - { 0}, and let Cut (p) be the cut locus of

pin M. We define the (n -1)-form dO" on M\ (Cut (p) U {p}) by

dO" ~ ( exp;^1 )* d0"3n-1,