1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

492 B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS

and similarly for a22 and a33. D

A nice property of the cross curvature tensor is the following result due

to Hamilton.

LEMMA B.18 (Bianchi-type identity for Cij). If (M^3 , g) has negative

sectional curvature (or positive sectional curvature), then c is a metric and

(B.32) ( c -l)ij \liCjk = 2 1 ( C -l)ij \7 kCij.

This implies id : (M^3 , c) ---+ (M^3 , g) is a harmonic map.

REMARK B.19. We may think of this result as dual to the fact that

if Re is positive (negative) definite, then the identity map id : (Mn, g) ---+

(Mn,± Re) is a harmonic map (Corollary 3.20 on p. 86 of Volume One).

PROOF. Using \liEij = 0 (which follows from the contracted second

Bianchi identity), we compute

(c-^1 )ij \licjk = (detE)-^1 Eij\li ( detE (E-^1 )jk)

• 1
  = ( det E)^1 \7 k det E =

2

\7 k log det c

= 1 ( 2 C-1) ij \lkCij,

where det c ~ det Cij / det 9ij.

Now given two Riemannian metrics c and g on a manifold M, the Lapla-

cian of the identity map id : (M, c) ---+ (M, g) is given by

(~id)k = (c-^1 )ij (rfj -~fj)

(B.33) = - (c-^1 )k£ (c-^1 )ij (\liCj£ + V'jCi£ - \leCij) = o,

where rt and f),.fj denote the Christoffel symbols of g and c, respectively.

Here we used (B .32). D

REMARK B.20. The above proof is the solution to Exercise 3.23 of Vol-

ume One.

3.2. The cross curvature flow and short-time existence. We say

that a 1-parameter family of 3-manifolds (M^3 ,g (t)) with negative sectional

curvature is a solution of the cross curvature flow (XCF) if

a

atg = 2c.

Likewise, if g (t) has positive sectional curvature, we say that (M^3 ,g (t)) is

a solution if

a

atg = -2c.

We have the following result due to Buckland [34].