282 A. THE RICCI CALCULUS
The Lie derivative is defined on vector fields so that
.CxY= [X,Y]for all X, YE C^00 (T Mn), where [X, Y] is the unique vector field such that
[X, Y] (f) = X (Y (!)) - Y (X (!))for all differentiable functions f : Mn ----t R If() is a covector field and X, Y
are vector fields, the Lie derivative of () is given by
(.Cxe) (Y) = x (() (Y)) - () ([X, Y]).
Note that
(.Cxe) (Y) = d() (X, Y) + y (() (X)))
where dis the exterior derivative defined in Section 3 below. Note too that
even though the Lie derivative is independent of any Riemannian metric g
on Mn, the Lie derivative of a vector field can be calculated with respect to
the covariant derivative V induced by g by the formula
(A.1) .CxY = [X, Y] = VxY - VyX.
Similarly, the Lie derivative of a covector field may by calculated by(.Cxe) (Y) = (V xe) (Y) +()(Vy X).
More generally, the Lie derivative of a (p, q)-tensor field A satisfies
(A.2a)
(A.2b)
(A.2c)
(A.2d)(.CxA) (Y1, ... , Yp; e1, ... , ()q)
= X (A (Y1, ... , Yp; e1, ... , ()q))
- 2=1~i~pA(Y1, ... ,[X,Yi] , ... ,Yp; e1, ... ,eq)
- l:i~j~qA (Y1, ... , Yp; e1, ... , .Cxej, ... , eq)
for all vector fields X and Y1, ... , Yp, and all covector fields e1,... , ()q· A
specific case of some interest for us is the observation that the Lie derivative
of the metric itself satisfies
(.Cxg) (Y, Z) = g (VyX, Z) + g (Y, V zX)
for all vector fields X, Y, Z. Hence in local coordinates, .Cxg is just the
symmetrized first covariant derivative of X, namely
(.Cxg)ij = (.Cxg) C:j~i, 0 ~j) = ViXj + VjXi.
A first-order differential operator of particular interest in Chapter 3 is
the divergence
fJ: C^00 (T* Mn ®s T* Mn) ----t C^00 (T* Mn)
defined for all symmetric (2, 0)-tensors A and all vector fields X by
n
(fJA) (X) = - (div A)(X) = L (VA) (ei, ei, X),
i=l