TD 3 du 29/02/2016

Mn(R)
06r6n n >2
det : A7→ det(A)C∞Mn(R)
det
SLn(R)C∞Mn(R)
n−1Mn(R)
U0Mn(R)A C
B D ∈U
Ir+A C
B D r D =B(Ir+A)−1C
Vr⊂Mn(R)r
Mn(R)
r
r
r(n−r)
6n−1Mn(R)
0
FRn
{A∈ Mn(R)|ker A=F} {A∈ Mn(R)|A=F}
Mn(R)
N G(k, n){A∈
Mn(R)|ker A∈N} {A∈ Mn(R)|A∈N} Mn(R)
MRnT:x7→ TxM
M G(n−1, n)
SUN
SLN(C)C∞MN(C)
Id
SLN(C)C∞MN(C)
Id UN

SLN(C)UNId
SUNC∞MN(C)
Id
G Gln(R)
G Gln(R)M∈ Mn(R) exp(M) =
∞
P
i=0
Mi
i!kMk<1L(Id +M) =
∞
P
i=0
(−1)iMi+1
i+1
L(A, B)∈ Mn(R)
(eA
keB k)k−→
k→∞
eA+B(eA
keB ke
−A
ke
−B
k)k2−→
k→∞
eAB−BA
LG={M∈ Mn(R)| ∀t∈R, etM ∈G} LG
Mn(R)Mn(R) (A, B)7→ [A, B] = AB −BA
G
Mn(R)
FLGMn(R)
ϕ:Mn(R) = LG×F→GLn(R)
(A, M)7→ eAeM
(Mk)k>0F\ {0}0
∀k>0, eMk∈G
U0LGV
InGLn(R)ϕ|UC∞U V ∩G
LGG In
LGG
M N f :M→N C∞
f f−1(f(x))
f(M)N
f
f−1(f(x)) f(M)N