1547671870-The_Ricci_Flow__Chow

280 A. THE RICCI CALCULUS

Although there is no natural isomorphism (in the functorial sense) be-
tween the bundles T Mn and T* Mn, the metric

g = 9ij dxi 0 dxj

and its inverse

g-^1 =lJ .. ~®~ a a

uX^1 uxJ

induce canonical isomorphisms Tj Mn ---+ r:: Mn whenever p+q = p' +q'. In
particular, the metric dual of a vector field XE C^00 (T Mn) is the covector
field Xb E C^00 (T* Mn) defined by

Xb = (Xigij) dxj;

and the metric dual of a covector field e E C^00 (T* Mn) is the vector field

e~ E C^00 (T Mn) defined by

a

e~ = (Bigij) oxF

In formulas involving the components of familiar tensors in local coordinates,
we will often use these isomorphisms without explicit mention, for instance
writing Rf for gjk ~j or Rke for gik gje ~j.

2. First-order differential operators on tensors
   Differentiation of functions on a smooth manifold is easy to define. If

X E C^00 (T Mn) is a vector field and f : Mn ---+IR is a differentiable function,

then X acts on f to produce the function X (!); in local coordinates

x (f) =xi ~f ..

ux^2

(Here and everywhere in this book, the Einstein summation convention is

in effect.)

To differentiate tensors requires a connection. It is a standard fact in

geometry that a connection in a vector bundle is equivalent to a definition

of parallel transport, which is in turn equivalent to a definition of covariant

differentiation. Thus in the present context, we may in particular identify

the Levi-Civita connection r of g with the covariant derivative, which is

the unique operator

\7: C^00 (T Mn)---+ C^00 (T Mn 0 T* Mn)

with the property that

X (g (Y, Z)) = X ( (Y, Z)) = g (\7 x Y, Z) + g (Y, \7 x Z)

for all vector fields X, Y, Z E C^00 (T Mn), where

'Vy X = (\7 X) (Y) E C^00 (T Mn).