286 A. THE RICCI CALCULUS
- Commuting covariant derivatives
Recall that the (3, 1)-Riemann curvature tensor is defined so that
(A.6) R (X, Y) Z = (V^2 Z) (X, Y) - (V^2 Z) (Y, X) E C^00 (T Mn)
for all vector fields X, Y, Z. Following the the notational convention of Sec-
tion 5 above, we write this as
(A.7) R(X, Y) Z = VxVyZ - VyVxZ.
One advantage of this formula (hence of the convention adopted in Section 5)
is that it makes the geometric significance of the Riemannian curvature clear
by demonstrating its fundamental role in commuting covariant derivatives.
By (A.1) and (A.5), one may also write equation (A.6) in the more familiar
form
R(X, Y) z = Vx (VyZ) - Vy (VxZ) - V [X,Y]z.
The reader is warned that this form is used without parentheses by those
authors who do not follow the convention we adopted in Section 5. In local
coordinates, all brackets [ 8 ~;, a~i] vanish identically, so that regardless of
which convention one adopts, one can write unambiguouslyR (a~i' a~j) a~k = Rfjk3~e = (vivj - vjvi) (a~k).
The commutator formulas for covariant derivatives implied by equation
(A.6) are collectively known as the Ricci identities. For example, if X is
a vector field, then in local coordinates,[Vi, Vj] Xe= ViVjXe - VjViXe = RfjkXk.
The commutator for a covector field () (equivalently, a 1-form) may be ob-
tained from this by using the metric dual ett' yielding
[Vi, Vj] ()k = ViVj()k - VjVi()k= 9ke [Vi, Vj] (ett)e = 9keRfjm (grmer) = - Rfjkee.More generally, if A is any (p, q)-tensor field, one has the commutator[V· i, v J ·] ACi-··Cq k1···kp = - V ·t V-AC1···Cq J k1···kp - v J ·V-AC1···Cq t k1···kp
q p= '°"" RCr AC1···Cr-1 mCr+1···Cq _'°""Rm AC1···Cq
L.....t t)m k1···kp L.....t t)ks k1···ks-l mks+1···kp ·
r = l s=l
Notes and commentary
The contents of this appendix are standard parts of classical Riemannian
geometry. For the convenience of the reader, we have reviewed them here in
order to establish an unambiguous notation.