1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

9. BUSCHER DUALITY TRANSFORMATION 47

and {ya:n=l be local coordinates on Mand P, respectively, such that ha/3 ~

h ( aia, a~f3) = 6af3· We then have

b9ij = 9ij, ttgij = 9ij,

b 9a:f3 = A6a:,e, ttga:,e = A-160:,e,

and the rest of the components are zero, where

b -b(f) f))

9ij ::;= g EJxi ' EJxj '

b -b(f) f))

g a,6 ::;= g EJya: ' oy.6 '

and similarly for tt g.
The explicit formulas below are from Haagensen [175] (see also (1.38) or
§J in Chapter 9 of Besse [27] for curvature formulas for warped products).

LEMMA 1.77. The Christoffel symbols of b g are

brfj = rfj,

b ria: /3 -- 60: (^131) :2 Y'i. log A,
b. A.
r~ 13 = -60:132"° \7i log A,
br~/3 = br~j = brij = o,
and likewise, the Christoffel symbols of the dual metric ttg are
ttr7j = rfj,
ttr~ ia: = -6/3 a: ~ 2 \7 · i log A '
tt. 1.
r~ 13 = 60:13 2 A V'i log A,
ttr'Y a:/3 -- ttri a:j -- ttr'?' ij -- o.
LEMMA 1. 78. The Ricci tensor of b g is given by
bRa:/3 = -~ [~logA+ ~ IV'logAl
2
] 60:13,
b Ra:i = 0,
b~j = Rij - ~\7i\7jlogA-~\7ilogA\7jlogA,
and the Ricci tensor of ttg is
ttRa:f3 = -
2
~ (-~logA + ~ IV'logAl
2
) 60:13,
ttR a:i · -O - '
tt q q

Rij = Rij + 2 \7 i \7 j log A - 4 \7 i log A \7 j log A.