- RIEMANNIAN GEOMETRY 447
(3) Contracted second Bianchi identity:. 1
(Vl-3.13) }J Rij =
2
"\liR,which is obtained from (Vl-3.18) by taking two traces (e.g., multi-
plying by gq.egjk and summing over q,£,j, k).1.3. Cartan structure equations. For metrics with symmetry, such
as rotationally symmetric metrics, it is convenient to calculate with respect
to a local orthonormal frame, also called a moving frame.
Let { ei} ~=l be a local orthonormal frame field in an open set U c Mn.
Denote the dual orthonormal basis ofT* M by { wi} ~=l so that g = L:~=l wi®
wi. The connection 1-forms w{ E 01 (U) are defined by(Vl-p. 106a)n
"\! xei ~ L w{ (X) ej,
j=l
for all i = 1, ... , n and X E C^00 (T Miu). They are antisymmetric: w{-wj. The first and second Cartan structure equations are
(Vl-p. 106b) dwi = wj /\ wj,
(Vl-p. 106c) Rm i^1 = n1 i = dw1 i - w~ i /\ WJk..
The following formula is useful for computing the connection 1-forms:
(A.1) wi k (eJ·) = dw i (ej, ek) + dwJ. (ei, ek) - dw k (ej, ei).
1.4. Curvature under conformal change of the metric. Let g
and g be two Riemannian metrics on a manifold Mn conformally related by
g = e^2 ug, where u : M -+ R If {ei}~=l is an orthonormal frame field for
g, then {ei}~=l, where ei = e-uei, is an orthonormal frame field for g. The
Ricci tensors of g and g are related by
(A) .2 R(-c e.e,ei -) =e -2u(Rc(e.e,ei)+(2-n)"\lee"\leiu-6.eib.u +l"\lul2(2-n)6i.e-(2-n)e.e(u)ei(u) ).
Tracing this, we see that the scalar curvatures of g and g are related by
(A.3) R = e-^2 u ( R - 2 (n - 1) flu - (n - 2) (n - 1) l"\lul^2 ).
For derivations of the formulas above, which are standard, see subsection
7.2 of Chapter 1 in [111] for example.
1.5. Variations and evolution equations of geometric quanti-
ties. The Ricci fl.ow is an evolution equation where the variation of the
metric (i.e., time-derivative of the metric) is minus twice the Ricci ten-
sor. More generally, we may consider arbitrary variations of the metric.
Given a variation of the metric, we recall the. corresponding variations of
the Levi-Civita connection and curvatures. (In Volume One, see Section 1
of Chapter 3 for the derivations, or see Lemma 6.5 on p. 174 for a summary.)