1547671870-The_Ricci_Flow__Chow

72 3. SHORT TIME EXISTENCE

of degree at most k + £ in (.
We now regard the Ricci tensor Re (g) as a nonlinear partial differential
operator on the metric g,

Reg= Re (g) : C^00 (StT* Mn) ---+ C^00 (S2T* Mn).

Here C^00 (StT* Mn) denotes the space of positive definite symmetric (2, 0)-

tensors (in other words, Riemannian metrics) and C^00 (S2T* Mn) denotes

the space of symmetric (2, 0)-tensors. By Lemma 3.5, the linearization

of the Ricci tensor is given by

1

[D (Reg) (h)]jk = 2,gpq (\lq'Vjhkp + V'q\lkhjp - V'q\lphjk - V'j\lkhqp).

Let ( E C^00 (T* Mn) be a covector. The principal symbol in the direction

( of the linear partial differential operator D (Reg) is the bundle homomor-

phism

a [D (Reg)](() : S2T* Mn---+ S2T* Mn

obtained by replacing the covariant derivative \Ji by the covector (i, namely

1 { (q(jhkp + (q(khjp }

[0-[D (Reg)](() (h)]jk = 2,gqp..

-(q(phjk - (1(khqp

(3.10)

A linear partial differential operator Lis said to be elliptic if its principal

symbol 0-[L] (()is an isomorphism whenever ( # 0. A nonlinear operator N

is said to be elliptic if its linearization D N is elliptic. There is a rich existence

and regularity theory for linear elliptic operators. However, as we shall see

below, the fact that the Ricci tensor is invariant under diffeomorphism,

(3.11) Re (<pg) = <p (Re (g)),

implies that the principal symbol 0-[D (Reg)](() of the nonlinear partial

differential operator Reg has a nontrivial kernel.

2.2. The Bianchi identities. To increase our understanding of the

consequences of the diffeomorphism invariance of the Riemann curvature

tensor, we shall show that it implies the first and second Bianchi identities.

We follow a proof of Kazdan [81], which is implicit in work of Hilbert. We

will see that the Bianchi identities are a consequence of the Jacobi identity

for the Lie bracket of vector fields, which in turn follows from the diffeomor-

phism invariance of the bracket.

Let X, Y, and Z be arbitrary vector fields on a manifold Mn. Let <pt

be the one-parameter group of diffeomorphisms generated by X such that

<po = idMn. Diffeomorphism invariance of the Lie bracket implies that