APPENDIX A
The Ricci calculus
From the viewpoint of Riemannian geometry, the Ricci flow is a very
natural evolution equation, because it can be formulated entirely in terms
of intrinsically-defined geometric quantities and thus enjoys full diffeomor-
phism invariance. Nevertheless, when studying the properties of the Ricci
flow as a PDE and in particular computing the evolution equations satisfied
by these intrinsic geometric quantities, it is often more convenient to use
(classical) local coordinates rather than (presumably more modern) invari-
ant notation. Consequently, we provide here a brief review of some aspects
of the Ricci calculus, concentrating on its application to covariant differen-
tiation in local coordinates.
1. Component representations of tensor fields
If Mn is a smooth manifold and 7r : £ ----+ Mn is any differentiable
vector bundle, standard constructions in geometry produce its dual bundle
£,various tensor products like£@[®[*, symmetric products like£ ®sE,
and exterior products like AP(£). Everything we shall say below extends
readily, mutatis mutandis, to this general situation. However, to keep the
exposition concise and concrete, we will specialize to the case that E = T Mn
is the tangent bundle of Mn.
Let (Mn, g) be a (smooth) Riemannian manifold. Then a (p, q )-tensor
A is a smooth section of the vector bundle
T%Mn ~(@PT* Mn) 0 (®qT Mn)over Mn. We denote this by writing A E C^00 (T3Mn). For example, a
vector field is a smooth section of TJ-Mn = T Mn, while a covector field
is a smooth section of Tf Mn = T* Mn. If U c:;;; Mn is an open set and
cp: U----+ ]Rn is a chart inducing a system of local coordinates (x^1 , ... , xn) on
U, we define the component representation ( A;:.-.-.·;Pq) of A with respect
to the chart cp : U ----+ ]Rn byA =A ki .. ·k .q dxJ.^1 ®···®dxP@J. --a ®···®--a.
Jl ""]p 8xk1 axkq
Equivalently,Ak1···kq J1 ... Jp =A(~8xJ1 , ... '~., 8xJP dxk1, ... ,dxkq).
279