2. FIRST-ORDER DIFFERENTIAL OPERATORS ON TENSORS 281In terms of the Christoffel symbols (rt) induced by the Levi-Ci vita con-
nection in the chart 'P : U ----+ lftn, the covariant derivative of a vector field X
is the (1, 1)-tensor with components
vixk = (VX) (
8~i) =
8~ixk + r~jxj.
In particular, it is customary in local coordinates to denote the operator
\7 q simply by \7 i.
axi
The Levi-Civita connection also defines a covariant derivative \7* for
covectors. This is the unique operator
\7* : C^00 (T* Mn) ___. C^00 (T* Mn© T* Mn)
with the property that
x (e (Y)) = ('VxB) (Y) + e (\7 x Y)
for all X, YE C^00 (T Mn) and e E C^00 (T* Mn), where'lxB = ('VB) (X) E C^00 (T Mn).
More generally, the Levi-Civita connection defines a covariant deriva-
tive "\J(p,q) on each tensor bundle TJ Mn. This is the first-order differential
operator
'V(p,q): C^00 (TJMn)----+ C^00 (r:+ 1 Mn)defined by the requirement thatX(A(Y1,.. .,Yp; B1,.. .,eq)) = (v~,q)A) (Y 1 ,.. .,Yp; B1,. .. ,eq)
p
+LA (Y1,.. ., \7 x Yi,.. ., Yp; B1,.. ., Bq)
i = l
q
+ 2:A(Y1,.. .,Yp; B1,.. .,'VxBj,. .. ,eq)j=l
for all (p, q)-tensors A, all vector fields Y 1 , ... , Yp, and all covector fields
B1, ... ,eq, where(\7~,q) A) (Y1, ... , Yp; B1, ... , Bq) = ('V(p,q) A) (X, Y1, ... , Yp; B1, ... , Bq) ER
Note that 'V(O,l) = \7 and \7(1,0) = \7*. The component representation of
the covariant derivative is defined by\7 i (p,q) Aki ···kq ....:... ( ) (^8 8 8. ki d k )1 ...^1. 1. P -=- \7 A ~, uxi uxJ1 ~, ... , uxJP ~, dx , ... , x q •
REMARK A.l. It is customary and harmless to abuse notation and de-
note all "\J(p,q) simply by \7. We adopt this convention throughout this
volume.