1547671870-The_Ricci_Flow__Chow

9. PROPERTIES OF THE NORMALIZED RICCI FLOW 217

(Note that we are now dropping the .9 ([) notation used above.) The easiest

way to obtain the evolution equations for quantities depending on g is by
first making a more general observation.

LEMMA 6.62. Suppose that g (t) is a smooth one-parameter family of

metrics on a man~fold Mn such that

a

at 9 = c.pg,

where c.p is a function of time alone.

(1) The Levi- Civita connection r of g is independent of time.

(2) The (3, l)-Riemann curvature tensor of g is independent of time.

(3) The ( 4, 0) -Riemann curvature tensor Rm of g evolves by

a

at Rijke = c.pRijkf.·

( 4) The Ricci tensor Re of g is independent of time.

(5) The scalar curvature R of g evolves by

a

at R = -c.pR.

PROOF. All the formulas are immediate consequences of Lemma 6.5

except for (3), which follows from (2) when we write

:t (Rijke) = :t (gcmRi'Jk) = c.pRijkf.·

D

REMARK 6 .63. The formulas above may also be deduced from Lemma

6.57 by observing how the various geometric quantities under consideration

respond to scaling the metric. (Compare with Lemma 17.1 of [58].)

COROLLARY 6.64. Let (Mn,g(t)) be a solution of the normalized Ricci

flow, and let r ( t) be defined by ( 6. 61).

(1) The Levi- Civita connection r of g evolves by

:tr7j = -\liRJ - YjR7 + vkRij·

(2) The (3, l)-Riemann curvature tensor evolves by

gt Rfjk = t::.Rfjk + gpq ( RijpR;qk - 2R;ikRJqr + 2R~irRJqk)

• Rf R~jk - R~Rlpk - R~Rfjp + R~Rfjk'
  (3) The (4, 0)-Riemann curvature t ensor Rm of g evolves by
  a
  at Ri.ikf. = f:::.Rijkf. + 2 (Bijkf. - Bijf.k - Bif.jk + BikjC)

+ RkJjRmi - R"!JiRm] + R'!JcRmk - Ri'JkRme + r Rijke,

where the tensor B is defined in ( 6.15).