1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. HEAT KERNEL ASYMPTOTICS FOR A TIME-DEPENDENT METRIC 281

REMARK 24.23. Below we shall use the following variant of (24.47).

Integrating (24.44), we have for s E (0, r 7 (x)]

(24.48)

where we used x~ (! (s)) = Tr sx(~) X and r 7 (! (s)) =sin the first line.

By applying the formulas (24.46) and (24.47) to (24.45), we have

Expanding this formula, we obtain the following.^3

LEMMA 24.24 (Asymptotic expansion for 'lj.;o).

Next we consider 'lj.;1. In view of (24.17) for k = 1, we first compute

flx,-r'lj.; 0. In general, consider the expansion in normal coordinates for the

Laplacian on a Riemannian manifold (Mn, g)

(^3) Compare with the asymptotic expansion (23.116) of c/Jo (x, y) for a fixed Riemannian
metric.