1547671870-The_Ricci_Flow__Chow

4. RELATION TO THE HARMONIC MAP FLOW 87

At the origin of a normal coordinate system for g, we have

Aij k = "2 1 ( Re -1) kf. ( V'iRje + Y'j~f. - V'e~j ).

Since both sides of this identity are components of tensors, it holds every-
where. Applying the contracted second Bianchi identity to the result of the
previous corollary, we get

(uA 9 ,h i "d)k - g ijAk ij - 2 1 (R c -1)kf. [ g ij (n·vi R· JC + n v J ·R· ie - n v e R· iJ ·)] -_^0.

D

REMARK 3.21. If Re < 0, then the result holds for the target manifold

(Mn, - Re (g)).

4.2. The cross curvature tensor. Corollary 3.20 raises some inter-

esting albeit tangential questions. We shall introduce these here only briefly,

because they are not needed for the remainder of this volume. However, we

plan to provide a more complete discussion in the next volume. (Also see

[ 32 ] for more details.)

PROBLEM 3.22. Given a Riemannian manifold (Mn,g), does there exist

a metric 'Y # g on Mn such that id : (Mn, 'Y) -+ (Mn, g) is a harmonic

map?

Assuming the sectional curvatures of g have a definite sign, the answer

in dimension three is given by the cross curvature tensor. If we choose a

local orthonormal frame in which the sectional curvatures are K;i ~ R2332,

K;2 ~ R 133 1 , and K;3 ~ R1221, then the Ricci tensor corresponds to the matrix

and the cross curvature tensor c = c (g) of (M^3 , g) corresponds to the

matrix

c = ( "'"

3

<1<3 <1<,). '

More explicitly, one can write c in terms of the Einstein tensor

1

E = Rc-- 2 Rg '

which has eigenvalues - K;1 , - K;2, and -K;3. We have

c ·. iJ = (det det E) g (E-1) ij = 2 ~ μ μ ipq jrs E E pr qs

_ - 8μ 1 pqk μ rse R· iCpq R kJr. S

where μijk are the components of the volume form with indices raised and

normalized so that μ^123 = 1 in an oriented orthonormal frame.