- PERELMAN'S FORMALISM IN POTENTIALLY INFINITE DIMENSIONS 419
- 1
\laJJj = \laiaj -IIijl/ = \laiaj - ( N)_ 112 Rijl/·
R+ 2T
By the Gauss equations, we have
(R(ai, aj)ak, ae)
= (R(ai,aj)ak,ae) - (n(ai,ae),n(aj,ak)) + (n(aj,ae),n(ai,ak))
1
= (R(ai, aj)ak, 8e) - ( N) (Ri£Rjk - Rj£Rik).
R+ 2T
By Koszul's formula for the Levi-Civita connection of g, i.e.,
2 (V x Y, Z) = X (Y, Z) + Y (X, Z) - Z (X, Y)
- 1
- ([X, Y], Z) - ([X, Z], Y) - ([Y, Z], X),
where the inner products are with respect tog, we have
- 1
(8.58) \1 aiv = N l/ 2 Re (8i),
(R + 27)
~ 1
(8.59) \1 vv = - ( N) \1 R.
2 R+ 2T
To derive the two formulas above, we used (Vaiv, v) = 0, (Vvv, v) = 0, and
2(Vaiv,8k) = v(8i,8k) = ( R+ ~)-l/
2
87 gik,(- ) \lkR
\lvv,ak = -([v,ak],v) = ( N)
2 R+ 2T
(these follow from Koszul's formula and (8.57)).
Applying another covariant derivative to (8.58) and (8.59), we have
(
- ) 1 1
\1 0 .\lvv,aj = 2 \liR\ljR- ( N)\li\ljR,
i 2(R+~) 2 R+2T
- ) 1 1
- (Vv fj aiv, 8j) = l N 2 (aTR -
2
N 2 ) Rij + l N (~eRej - 8TRij),
2 ( R + 2T) T R + 2T- (V[a· v]v, 8j) = - l 2 \liR\ljR,
'' 4(R+ ~)
where, to obtain the second formula, we used
(Var (Rc(ai)) ,aj) = aT~j - (Rc(8i) 'Varaj)
= 8TRij - Ri£R£j,
(note that (Varaj,ak) = ~8 7 (8j,8k) = Rjk), and the third formula follows
from (8.57) and (8.59).
Hence the curvatures in the normal direction are given by