1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. DIFFERENTIATING A CONVOLUTION WITH THE PARAMETRIX 245

so that
8 xi (z)
-;::;-rd UXz (x, z) = d ( X,Z )"

On the other hand, consider the unit speed minimal geodesic

(3: [O, d (x, z)] -+ M

from z to x. The unit tangent vector of (3 at x (and any other point along
(3) is

· ~ xJ (z) 8

(3 (d(x,z)) = -L.t. 1 d( x, z ) ux ~ J"

J=

Thus, by the. first variation of arc length formula

a /. a)

ax~ d (x, z) = \(3 (d (x, z))' axi x

/ ~ xJ (z) 8 8 )

= \ -ki d(x,z) 8xJ' 8xi x

xi(z)

d(x,z)'

so that

~

8

. d^2 ( x' z) = 2d ( x' z) ~

8

ux; ux;. d ( x' z)

= -2xi (z).

Clearly (23.98) follows.

Next we claim that we have

~

2

.d^2 (x,z)- ~

2

.d^2 (x,z)=O(d^2 (x,z)).

U ~Xi XU ~XJ X OXi z OXJ Z

(23.99)

To see this, note that

32 2 8. )

.. d (x, z) = ~ i (2xJ (z) = 20ij·
ax1ax{ UXz

On the other hand, since {xi} ~=l are geodesic normal coordinates centered

at x, we have rfj (x) = 0. Letting ei = a~i (x), so that {ei}~=l is an

orthonormal frame at x, we compute

(23.100)

Indeed,