Corrigé 05 Fichier

γ(t)=(t, c) = t+ic Xt=a1(t) + ia2(t)
γ X
∇tXt= 0
˙a1
˙a2=01
c
−1
c0·a1
a2
Xt=C1sin(t/c) + C2cos(t/c) + i(D1sin(t/c) + D2cos(t/c))
R3
R2
R2
B:TgG−→ g
x7−→ X(p) = dgLpx
A X
S2
TeG
Lg∗[X, Y ] = [Lg∗X, Lg∗Y]
Lg

n(R)n(R)
TeGLn(R)=Mn(R).
n(R)n(R)
SLn(R) = {M∈Mn(R)|detM−1=0}.
f
f: Mn(R)−→ R
M7−→ detM−1
n(R)
fn(R)
dIf·H= trH.
n(R)I
A B AB −BA
X Y θt
M Y
[X, Y ] = d
dt|t=0θt∗X.
B∈n(R) = TIn(R)
Bg) = gB
θt(g) = gexp(tB).
θt∗A= exp(tB)Aexp(−tB).
0BA −AB
s
fs:t7→ γ(s+t)gs:t7→ γ(s)γ(t).
s f g
fs
∇˙gs˙gs
gs(t) = Lγ(s)γ(t).
˙gs(t) = dγ(t)Lγ(s)˙γ(t).
∇
∇˙gs(t)˙gs(t) = Lg∗∇˙γ(t)˙ γ(t)= 0.

fsgs
fs(0) = gs(0) = γ(s).
ϕ(s) = ˙ fs(0) ψ(s) = ˙gs(0).
ϕ ψ
0 ˙γ(0) γ(0) = e
∇ϕϕ=∇ψψ= 0
n