1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

420 8. APPLICATIONS OF THE REDUCED DISTANCE

(R(oi,v)v,aj) =

1

2 [(a7R- 2 N 2 ) Rij + lviRY'jRJ

2 (R+ ~) T

• (R~ ~) (-a7Rij -t\7i\7jR+RieRej)'

(R(8i,8j)ok,v) = (Va/::J&j8k-V&jV&i8k,v)

1.

312 (Rjk \i'iR - Rik \ljR)

2 (R+ ~)

1

N 1/2 (\i'iRjk - Y'jRik).

(R + 27)

In terms of Oi and 8n this says

(R(8i,8- 7 )8n8j) =^1 [( N)·^1 ]

2

(R+ ~) 87R- 272 Rij + 2,\i'iR· V'jR

• ( RieRej - t \7i\7jR-87Rij) ,

(R(oi,aj)ak,a7)= (

1

N)(RjkY'iR-RikvjR)-(ViRjk-V'jRik)·

2 R+ 27

Note that the curvatures of g are the components of Hamilton's matrix

Harnack expression in the following sense:

(R(8i, 8j)8k, 8e) = Rijke mod O(N-^1 ),

( R(8i,87)8n8j - ) = -87Rij-^1 1 -1

2

viY'jR+RieRej -

27

Rij mod O(N )

1

= ~Rij - 2 V'iY'jR + 2Rikj£Rke

• RieRej -^1 1
  27

Rij mod O(N-),

(R(8i,8j)8k,87) = \i'iRjk - Y'jRik mod O(N-^1 ).

Taking the trace gives the entries of the trace Harnack expression:

• Rc(8i, 8j) -= Rij mod O(N -1 ), ..


• _ 1 -1
  Rc(oi,87) =

2

vjR mod O(N ),

Rc(8n 87) = t ~~ = t.6.R + 1Rcl^2 mod O(N-^1 ).

Now consider (M, g) as a warped product:

M =B Xp F,

where the base manifold is (B, 9B) = (M, g), the fiber is (F, gp) =(SN, 9af3),

and f =ft. We use O'Neill's formula to compute the curvatures of (M, g).