1547671870-The_Ricci_Flow__Chow

68 3. SHORT TIME EXISTENCE

Hence

D

LEMMA 3.2. The variation of the Levi-Civita connection r is given by

(3.3) atrij a k = Ike )

2 g Cvihje + vjhie - Vehij.

PROOF. Recall that

rij k = 2g^1 k£ (oigje + ojgie - oegij)

in local coordinates {xi}, where Oi ~ a~i. Hence

a k 1a k£

otrij = 2, otg · (oigje + ojgie - oegij)

• ~gk£ (ai (%tgje) + aj (%tgie) - ae (%tgij)).

In geodesic coordinates centered at p E Mn, one has rfj (p) = 0. It follows

that OiAjk = '! iAjk at p for any tensor A; in particular, Oigjk (p) = 0 for all

i, j, k. Thus we obtain

a k 1 ke ( a. a a )

at rij (p) = 2g vi atgje + v j atgie - v e at^9 ij (p) ·

Since both sides of this equation are components of tensors, the result holds

in any coordinate system and at any point. D

Since the Riemann curvature tensor is defined solely in terms of the

Levi-Civita connection, we can readily compute its evolution.

LEMMA 3.3. The evolution of the Riemann curvature tensor Rm is given

by

a 1 { vivjhkp + vivkhjp - vivphjk }

(3.4) ot Rfjk = 2iP.

• '! j '! ihkp - '! j '! khip + '! j '! phik

PROOF. In local coordinates {xi}, we have the standard formula

Rfjk = air;k - ajrfk + r~krfp -rfkr;v

Thus we compute