1547671870-The_Ricci_Flow__Chow

184 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE

whence

(Rm (U))ij = "'(Uij - Uji) = 211,Uij·

Now we can square the operator Rm to obtain the operator

Rm^2 = Rm o Rm : /\^2 T* Mn ~ /\^2 T* Mn

given in local coordinates by

(6.23) (Rm^2 )ijk£ = gPqgrs RijpsRrqkf·

REMARK 6.23. An equivalent theory results if one regards Rm and

Rm^2 as symmetric bilinear forms on /\^2 T Mn, hence as smooth sections

of /\^2 T* Mn @s /\^2 T* Mn. This point of view will be useful below.

Although Rm^2 is the most natural definition of the square of the Rie-

mann curvature operator, there is another concept of square which will be

useful. This can be defined whenever one has a Lie algebra g endowed with

an inner product (-,-). Choose a basis { <p°'} of g and let c~f3 denote the

structure constants defined by

[ <p°', <pf3] ~ I: c~f3 <p 'Y,

'Y

where [-, ·] is the Lie bracket of g. Let { <p~} denote the basis algebraically

dual to { <p°'}, so that <p~ ( <pf3) = 8~. Given a symmetric bilinear form L on

g*, we may regard L as the element of g ®s g whose components are given

by

Laf3 ~ L (<p~, <p~).

Then there is a commutative bilinear operation # defined for all L, M E

g @s g by

(L#M)af3 ~ C~f;C~( LryoME:(·

Abusing notation slightly, we define the Lie algebra square £# E g ®s g

of L by.

(6.24) (L#)af3 ~ (L#L)af3 = C~°C~( LryE:Loc·

The following property will be needed in showing that positivity of the

Riemann curvature operator is preserved by the Ricci flow.

LEMMA 6.24. If L 2: 0, then £# 2: 0.

PROOF. Without loss of generality, we may choose a basis { <p°'} that

diagonalizes L, so that Laf3 = 8af3Laa· Then for any v = v°'<p~ E g*, we

have

£# (v,v) = (v°'ct) (vf3c~c) LryE:Loc = (v°'C~^0 )2 LryryLoo.

D

REMARK 6.25. The inner product (-, ·) of g defines an metric isomor-

phism g ~ g* by v f-+ (v,-). This isomorphism allows us to regard L: g ~ g

as a self-adjoint endomorphism.