1547671870-The_Ricci_Flow__Chow

1. A CONFORMAL CHANGE OF METRIC 107

FIGURE 1. A 2-sphere whose curvature is of mixed sign

In the case of a surface M^2 , we have dw^1 = w^2 /\ w~, dw^2 = w^1 /\ wi, and

Rm§= dw~.

In particular, the Gauss curvature is given by

K ~ (R (e1, e2) e2, e1) = Rm§ (e1, e2).

LEMMA 5.3. If g and h are metrics on a surface M^2 conformally related

by

g = e2uh,

then their scalar curvatures are related by

Rg = e-^2 u (- 2b.hu +Rh)·

PROOF. Let {f 1, h} be an orthonormal frame field for the metric h. Let

{ T/^1 , T/^2 } be its dual coframe field, and denote the corresponding connection

1-form by T/~. Then

e1 =o=. e -uf 1 e2 =o=. e -uf 2