THÈSE Koléhè Abdoulaye COULIBALY-PASQUIER

Consequently:d(df(T − t, .) X Tt (x)(W T 0,tv))dM≡ −ddt df(T − t, .)(WT 0,tv) dt− 1 2 〈∇T −t f(T − t, .), ∂ t (g(T − t)) #T −t (W T 0,tv)〉 T −t dt+ 1 2 ∆ T −tdf(T − t, .)(W T 0,tv) dt− 1 2 〈( ˜df(T − t, U T t )), (U T t ) −1 (Ric g(T −t) −∂ t (g(T − t))) #g(T −t) (W T 0,t)v dt〉 R ndM≡ −ddt df(T − t, .)(WT 0,tv) dt + 1 2 ∆ T −tdf(T − t, .)(W T 0,tv) dt− 1 2−t)df(T − t, Ric#g(Tg(T −t)(W T 0,tv) dt.But recall that f is a solution of:so that∂∂t f = 1 2 ∆ tf,− ∂ ∂t df(T − t, .) = −1 2 d∆ T −tf(T − t, .).We shall use the Hodge-de Rham Laplacian □ T −t = −(dδ T −t + δ T −t d) which commuteswith the de Rham differential, and we shall use the well-known Weitzenböckformula ([16, 17]), which says that for θ a 1-form, □ T −t θ = ∆ T −t θ − Ric T −t θ. Weget:d∆ T −t f(T − t, .) = d□ T −t f(T − t, .)= □ T −t df(T − t, .)= ∆ T −t df(T − t, .) − Ric T −t df(T − t, .).Finally:d(df(T − t, .) X Tt (x)(W T 0,tv))dM≡1dM≡ 0,Ric 2 T −t df(T − t, .)(W T 0,tv) dt− 1 2 〈∇T −t #T −tf(T − t, .), Ric (W T 0,tv)〉 T −t dtT −tby duality; for a 1-form θ and for v ∈ T M:where〈θ # , v〉 = θ(v) .Ric(θ)(v) = Ric(θ # , v.)4914