1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

If () is a 1-form, then

(Vl-p. 286b)

1. RIEMANNIAN GEOMETRY

More generally, if A is any (p, q)-tensor field, one has the commutator

(Vl-p. 286)

[V'· tl V'·]A.e1···£q J ki···kp -'-\i'·V'·A.e1···£q -;- i J ki···kp -\i'·\i'·A.e1···£q J i ki···kp

q p

449

= ~ R4-r: Afr··£r-1 m.er+1···£q - ~ R'm: A.e1···£q

L.J iJm ki ···kp L.J iJks ki ···ks-1 m ks+1 ···kp ·

r=l s=l

1. 
   
   
   
   7. Lie derivative. Because of the diffeomorphism invariance of the
      Ricci fl.ow, the effect of infinitesimal diffeomorphisms on tensors, e.g., the
      Lie derivative, enters the Ricci fl.ow. (Seep. 282 in Section 2 of Appendix A
      in Volume One.)
      The Lie derivative of the metric satisfies
      (Vl-p. 282a) (.Cxg) (Y, Z) = g (\i'y X, Z) + g (Y, V' zX)
   
   
   
   

for all vector fields X, Y, Z. In local coordinates

(Vl-p. 282b) (.Cxg)ij = (.Cxg) (a~i' a~j) = Y'iXj + Y'jXi.

In particular, if X = V' 1 is a gradient vector field, then

(.C\JJ9)ij = 2\i'iV'jf.

1.8. Bochner formulas. (See p. 284 in Section 4 of Appendix A in
Volume One.) The rough Laplacian denotes the operators

~:coo (TiMn) ---+coo (TiMn)'

where T$ M ~ Q9P T* M ® ®q TM, defined by

(Vl-p. 284a)

n

(~A) (Yi, ... , Yp; B1, ... , Bq) = L (V'^2 A) (ei, ei, Yi, ... , Yp; B1, ... , Bq)

i=l

for all (p, q)-tensors A, all vector fields Yi, ... , Yp, and all covector fields
() 1 , ... , Bq, where { ei}~=l is a (local) orthonormal frame field. The Hodge-
de Rham Laplacian -~a: OP (M)---+ OP (M) is defined by

(Vl-p. 284b) -~a ~ d8 + 8d.

In particular, if() is a 1-form, then

(Vl-p. 284c) ~aB = ~() - Re (B).

For any function 1 : M ---+ JR

(A.4) ~V'1=V'~1+Rc(V'1)

and

(A.5) ~IV' 112 = 2IV'V'11^2 + 2 Re (V' 1, V' 1) + 2\i' 1 · V' (~1),