428 8. APPLICATIONS OF THE REDUCED DISTANCEwhere V'm is the covariant derivative associated to the metric grJ on M. The
other components of gm are given bygbQ = 9b0 - IV' 11
2= N - 2(~f-IY'fl^2 + R-~) - t -IY'fl^2 + 0 (N-^1 )
2r 2r r= ~ ( ~ - [r(2~f-IY'fl^2 + R) + f-n]) + O (N-^1 ),
g7/:p = 97/:p = ( 1 _· ~) 9ap,giQ = -Y'(1/J,.Moi)f + 0 (N-^1 ),
g:O = g: = 0.
PROOF. We haveHence
orgij f} m _ - or f} (•!•* '//7g ) ij + 0( N -1)
=('I/;;(~~) )ij + (£(1/J,.)*(\lf) (1/;;g))ij +O(N-1)
='l/;;(2Rc(g))ij +2V'iY'jfo'i! mod O(N-^1 )
= 2Rc(gmlMx{y}x{;})ij + 2V'iY'jf o 'it mod O(N-^1 ).We computegbQ = _gm ( ~ ( 07) ' ~ ( 87))
=_gm ((V' f, 0, 1), (V' f, 0, 1))
= IY'fl
2- 21v11^2 + 9b0 ·
= 9b0 - IY'fl
2
.