1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

430 8. APPLICATIONS OF THE REDUCED DISTANCE

where I'fj(M, g) and I'~.a(SN) denote the Christoffel symbols of (M, g) and

SN, respectively.

PROOF. Recall that the induced metric gm~ 9ml1ic on He is given by

9i'J (x, y, r) = 9ij (x, y, r) + 0 (N-^1 ) = gij (x, r) + 0 (N-^1 ),

ga,B -m( x,y,r ) = (i - 2f(x,r))-N ga,B ( x,y,r ) ,

9: (x,y,r) = 0.

We compute the Christoffel symbols f'~b as follows (where p denotes an index
in the M and sN directions, i, j, k, .e denote indices in the M direction only,
and a, (3, --y, 5 denote indices in the SN direction only):

f'k ij - 2 l(-m)kP([)·-m g igjp + (^0) Jgip .-m - [) pgij -m)
= 2 1 (-m)k£(£J g uigje -m + ujgie £J -m - uegij £J -m)
= I'fj(M,g) mod 0 (N-^1 ),
f'a ij - ~(-m)aP(o·-m 2 g igjp + (^0) Jgip .-m {) pgij -m)
= 2 1 (-m)a8(£J g Uigj8 -m + Ujgi8 £J -m -. u5gij £J -m)
=0,
f'a i,B - 2 1 (-m)aP({) g ig,Bp -m + (^0) ,Bgip -m - upgi,B £J - )

- _ l 2g _a'Y({) ig,8/ -m + U,Bgif £J -m - [) 'Ygi,8 -m)

La,!:} -m ^8 $\Jd 2

• 2g uig,8 1 = -~ mod 0 (N-),

f'k ia -_ 2 1 (-m)kP({) g igap -m + {) agip -m - {). pgia -m)

= 2 1 (g -m ) ( k£ aigae -m +Bagi£ -m - 8egia) -m

=0,

f'k a,B _ - 2 1 (-m)kP(f) g ag,Bp -m + (^0) .Bgap -m - upga,B £J -m )

= 2 1 (-m)k£(& g ag,Be -m +^0 .Bgae -m -^0 ega,B -m )

\Jkf

= rga,BN mod 0 (N-^2 ),