- SUPPLEMENTARY MATERIAL: ELEMENTARY TOOLS 261
so that if Mis given by (23.138), then
dd: 3 1s=O <let M =tr (A) ( tr
2
(A) - 3 tr ( (A)2) + 6 tr (B))
+ 2 tr ( (A)^3 ) - 6 tr (AB) + 6 tr ( C).
Finally, assuming dd~ (0) =A= 0, we have (where the RHS is evaluated at
s = 0)d d4 I
s^4 s=O detM
= 3 tr (M-1 d2 M) tr (M-1 d2 M) - 3 tr (M-1 d2 M M-1 d2 M)
ds^2 ds^2 ds2 ds2(
+tr M-1 d4M) ds4= 12tr^2 (B) - 12tr (B^2 ) + 24tr (D).The lemma now follows from<let M ( s) = <let M ( 0) + s d d I <let M + -^82 d d2 I
(^8) s=O 2 8 2 s=O <let M
83 d3 I 84 d4 I
+ 6 ds3 s=O detM + 24 ds4 s=O detM + 0 (s5).
D5.2. Interchanging differentiation and integration.
We recall an elementary fact about interchanging differentiation and
integration. The following is Theorem 13 on p. 297 of Widder [187].
LEMMA 23.38 (Fubini's theorem on noncompact space-time). Let (Mn, g)
be a noncompact Riemannian manifold. If(i) f (x, t) E C^1 (M x [a, w]),
(ii) the improper integral JM f (x, t) dμ (x) converges uniformly int E
[a,w],
then1w JM f (x, t) dμ (x) dt =JM 1w f (x, t) dtdμ (x).
REMARK 23.39. By assumption (ii), we have that for every 0 EM and
c: > 0, there exists Re < oo independent oft E [a, w] such that for every
RE [R:,oo),
r f (x, t) dμ (x) < c:.
lM-B(O,R)