L`exponentielle de SOn(R) est surjective

SOn(R)
son(R)→SOn(R)
son(R)
n= 2
son(R)gln(R)=Mn(R)
SOn(R) On(R)Sn⊂Mn(R)
f: Mn(R)→Snf(M) = tMM −Id
On(R) = {M∈Mn(R), f(M)=0}
fson(R) =
ker(dIdf)f(Id +H) = (Id +tH)(Id +H)−Id = tH+H+tHH dIdf(H) =
tH+H dIdf: Mn(R)→SnA∈Sn(1/2)A f
son(R) = ker(dIdf) =
son(R)H∈Mn(R)
exp(uH)∈On(R)u∈Ru t
texp(uH) exp(uH) = exp(utH) exp(uH) = Id ,∀u∈R.
1uId +u(tH+H) = Id
tH+H= 0 
RA x ∈AexpA(x)
Pn≥0xn/n!
f:A→B
x∈A f(expA(x)) = expB(f(x))
f f(PN
n=0 xn/n!) = PN n=0 f(x)n/n!
f
n= 2 SO2(R)
Rθ=cos(θ)−sin(θ)
sin(θ) cos(θ).

I=0−1
1 0 ∈so2(R).
I2=−1f(a+bi) = a+bI a a Id
Rf:C→M2(R)
exp(θI) = exp(f(θi)) = f(expC(θi)) = f(cos(θ) + isin(θ)) = cos(θ) + sin(θ)I=Rθ.
n
M∈SOn(R)P∈On(R)P MP −1
diag(Idr, Rθ1, . . . , Rθs).
n= 2
diag(0r, θ1I, . . . , θsI),
M
P−1diag(0r, θ1I, . . . , θsI)P .
P
SOn(R)
son(R)
son(R)f: Mn(R)→Sn
dIdf Sn
Mn(R)