l. VARIATION FORMULAS 69As in Lemma 3.2, we use geodesic coordinates centered at p E Mn to cal-
culate that
gt Rf 7 k (p) = vi (gt r]k) (p) - v j (gt rfk) (p) ,
and then observe that this formula holds everywhere. The present lemma
follows from substituting (3.3) into this equation. D
REMARK 3.4. By commuting derivatives, one can also write the evolu-
tion of the Riemann tensor in the form(^8) R e _^1 ep { v ivkhjp+vjvphik-vivphjk-vjvkhip}
at ijk - 29 q q ·
-Rijkhqp - Rijphkq
LEMMA 3.5. The evolution of the Ricci tensor Re is given by:
a 1
(3.5) 7'lRjk = -gpq (\7q\7jhkp + \7q\7khjp - \7q\7phjk - \7j\7khqp).
ut 2
PROOF. This follows from contracting on i =£in Lemma 3.3. D
REMARK 3.6. Recall that the divergence (3.19) of a (2, 0)-tensor is given
by
(8hh = - (div hh = -gij\7ihjk,
and denote the Lichnerowicz Laplacian [93] of a (2, 0) -tensor by
(3.6) ( !::iLh) jk ~ !::ihjk + 2gqp R~jk hrp - gqp Rjp hqk - gqp Rkp hjq·
(The Lichnerowicz Laplacian is discussed in Section 4 of Appendix A). De-
noting the trace of h by
H ~tr 9 h = gPqhpq,
one can write the evolution of the Ricci tensor in the form
gtRjk = -~ [!::iLhjk + \7j\7kH + Vj (8hh + \7k (oh)j].
LEMMA 3.7. The evolution of the scalar curvature function R is given
by:
(3.7) gtR = -/::iH + \7p\7qhpq - (h,Rc).
PROOF. Using Lemmas 3.1 and 3.5, we compute that
gt R = ( gtgjk) Rjk + gjk (gt Rjk)
= -lJg · · ke (\7i\7jhke - \7i\7khje + hikRje).
D
REMARK 3.8. One can also write the evolution of the scalar curvature
in the invariant form