160 J -M BISMUT AND W ZHANG GAFA

Proposition 2.6. The 1-form вд{Е, h^) is closed and its cobomology class does not depend on h^. More precisely, ifh'^ is another G-invariant metric, then

eg { F , h^^ ) - ^eg { F , h^ ) =d ( l0Jl |p-^(^-^^») ig)) . (2.16)

V V II lldet(F,{^}) / /

Also

/ 11 II'

log -^-^(^-^^»1 ig) = TT

. . o . |^

( 2 . 17 )

\ll \\det(F,{g}) ^

Proof : Our proposition follows from the considerations in (2.9)-(2.15). о Clearly M g is a totally geodesic submanifold of M. Let h^^^ be the

Riemannian metric induced by h^^ on TMg. Let V^^» be the Levi-Civita

connection on {TMg^h^^^),

Let e{TMg, V^^^) be the Chern-Weil representative of the rational Eu-

ler class of TMg, associated to the metric preserving connection V^^».

Then

e { TMg , V'^^ ) = Pf

27Г

if dim Ma is even ,

^ (2.18)

0 if dimM^ is odd .

Let h^^^ be another G-invariant metric and let V'^^^ be the sponding Levi-Civita connection on TMg.

Let е(ГМ^, V^^^ V'^^^) be the Chern-Simons class of dimM^ ~ 1 forms on Mg, such that

de { TMg , V^^^ , V'^^^ ) = e{TMg,V^^^)-e{TMg,V^^^) . (2.19)

Let now h^^, h^ and h'^^, /г'^ be two couples of G-invariant metrics on

TM , F . Let II II Дя(м,F),G) ^^^ II ir^f(H(M,F),G) be the corresponding

equivariant Ray-Singer metrics on det(iî(M, F),G).

We now give the following extension of the anomaly formula of [BZ,

Theorem 4.7].

THEOREM 2.7. For g EG, the following identity holds

tos ( " „ "'ff""'"-"''°')^'')= (2.20)

\ II \\det(H*iM,F),G) )

J

Jm

\ ! l lldet(F,{s})/

- / вд{Г, h^)è{TMg, V^^' , V™» ) .

Jm ,