422 8. APPLICATIONS OF THE REDUCED DISTANCE
By (2) we haveand hence
- 1
R(ai, Ba)aj = - JT Hess_g( vr)(ai, aj)Ba,
- R(ai,Ba)ar = -JTHess_g(yfT)(ai,ar)Ba,
- 1
R( an Ba)ar = -JT Hess_g( yfT)( an ar )Ba,
(il(ai,ea)aj,e(3) = 2 (R~f:r)~iga(3= 0 mod O(N-^1 ),
( R(ai, Ba)an e(3) = - 4 (R ~ f:r) 9a(3\liR
= 0 mod O(N-^1 ),
( R(ar, Ba)an ef3) = - 4 (R ~ f:r) ( arR + ~) gaf3
= 0 mod O(N-^1 ).
By (5) we haveand hence
R(ai, aj)ak = R(ai, aj)aki
R(ai,ar)ar = R(ai,ar)an( R(ai, aj)ak, ae) = \R(ai, aj)ak, ae)
1
= (R(ai, aj )ak, ae) - N (~eRjk - Rje~k)
R+ 2r= (R(ai, aj)ak, ae) mod O(N-^1 ),
(il(ai,aj)ak,ar) = \R(ai,aj)ak,ar)
1
2 (R + ~) (Rjk \liR - Rik VjR) - (\liRjk - Vj~k)