446 A. BASIC RICCI FLOW THEORY• trg denotes the trace with respect tog (e.g., of a symmetric (2, 0)-
tensor).- sn usually denotes the unit n-sphere.
- For tensors A and B, A * B denotes a linear combination of con-
tractions of the tensor product of A and B. - ~ denotes an equality which holds on gradient Ricci solitons.
- ~ denotes an equality which holds on expanding gradient Ricci
solitons.
1.2. Basic Riemannian geometry formulas in local coordinates.
In Ricci fl.ow, where the metric is time-dependent, it is convenient to compute
in a local coordinate system.
Let (Mn, g) be an n-dimensional Riemannian manifold. Almost ev-erywhere we shall assume the metric g is complete. Let {xi} be a local
coordinate system and let ai ~ 8 ~i • The components of the metric are
9ij ~ g (8i, 8j). The Christoffel symbols for the Levi-Civita connection, de-
fined by \1 8 iaj ~ I'fjak, are
(Vl-p. 24) rij k = 2,g^1 k/l, ( 8i9je + 8j9ie - 8egij),
where (gij) is the inverse matrix of (9ij). The components of the Riemann
curvature (3, 1)-tensor, defined by
(Vl-p. 286) R (8xi' a 8xj a) 8xk a.ea (^7) Rijk axe'
are
(Vl-p. 68)
The Ricci tensor is given by
(Vl-p. 92) R-iJ. -- RP pij -- Up !) rP ij - a-rp i pj + rq ij rp pq - rq pj rP iq·
The scalar curvature is R = gij ~j. If M is oriented and the local coordinates
{xi} ~=l have positive orientation, then the volume form is
(Vl-p. 70) dμ = yldetg dx^1 /\ · · · /\ dxn.
The Bianchi identities.(1) First Bianchi identity:
(Vl-3.17) 0 = ~jke + Rik£j + ~ejk·
(2) Second Bianchi identity:(Vl-3.18)
where we take \1 Rm, cyclically permute the first three indices of
\JqRijke (components), and sum to get zero.