1547671870-The_Ricci_Flow__Chow

284 A. THE RICCI CALCULUS

for all e E OP (Mn) and 'f/ E OP+^1. By integrating by parts, it is easy to
verify that
n

i=l

for all vector fields X 1 , ... , Xp, where { ei}~=l is a (local) orthonormal frame

field. Note that this formula can be used to define 8 even when Mn is not
compact.

4. Second-order differential operators
   A number of second-order differential operators are of considerable im-
   portance to our study of the Ricci flow.

The rough Laplacian denotes the family of operators

~: c= (TJMn) __, c= (TJMn)

defined by

n

(~A) (Y1, ... 'Yp; e1, ... 'eq) = L (\7^2 A) (ei, ei, Y1, ... 'Yp; e1,... 'eq)

i=l

for all (p, q)-tensors A, all vector fields Y 1 , ... , Yp, and all covector fields

e 1 , ... , e 9 , where { ei} ~ 1 is a (local) orthonormal frame field. In local coor-

dinates, we write

~AJ. k1 ·... ··J· kq ....:... -=-( ~A ) ( ~, {) ~' {) ... , ~, {). dx k1 , ... , dx k q ) •

(^1) P uxi uxJ (^1) uxJp
The Hodge-de Rham Laplacian (frequently called the Laplace-
Beltrami operator) denotes the family of maps
-~d: OP (Mn)__, OP (Mn)
defined by

• ~d = d8 + 88 = dp-l8p + 8p+ldp,
  where d and 8 are given in Section 3 above. It differs from the rough Lapla-
  cian by curvature terms which depend on the index p of the space OP (Mn).
  For instance, if e is a 1-form (equivalently, a covector field), then the com-
  ponents of ~de in local coordinates are given by the formula

~de= ( ~ei - R;ej) dxi.

If o: is a 2-form, then

(A.3) ~do:= ( ~O:ij + lPl°^9 Rijk.f.O:qp -l.e.~kO:.f.j - gkeRjkO:ie) dxi /\ dxJ.

Finally, the Lichnerowicz Laplacian, introduced in [93], is an operator

~L: C^00 (T* Mn ®s T* Mn)__, C^00 (T* Mn ® s T* Mn)