Fiche résumée du cours de géométrie différentielle 1

E U
R×E f :U→E f(t, x)C1
I⊂Ry:I→E C1
∀t∈I, (t, Y (t)) y0=f(t, y(t)) .
(y1, I1)>(y2, I2)I2⊂I1
y1|I2=y2
f t
x
t∀(t0, x0)∈U ut0,x0:
It0,x0→E ut0,x0(t0) = x0Cr+1 f
Cr
D={(s, t, x)∈R×R×E, s ∈It,x}R:D→
E R(s, t, x) = ut,x(s)D R f
∂f
∂x Cr−1R Cr
∂R
∂x t x It,x →
End(E)
z0(s) = ∂f
∂x (s, R(s, t, x)) = z(s), z(t0) = IdE.
f
x u :I→E
(b, x) (t, u(t)) t→b
I(b, x)/∈U U =V×R
b= sup I < ∞ ∀k⊂V∃c < b u(t)/∈K
c<t<b ku(t)k→ ∞ b
f x
I
f(t, x) = f(x)t
Av B
2A−λI B −λI E1,2
A, B C C
R
y0=f(t, y)f Cr+1 D⊂R×R×E
R:D→E Us,t ={x∈E, s, t, x ∈
D}
ϕs,t :x→Rs,t,x .

ϕs,t Us,t Ut,s
ϕr,s ◦ϕs,t =ϕr,t .
y0=f(y)Ut={x, (t, 0, x)∈D}
ϕt=ϕt,0ϕt,s =ϕt◦ϕ−1
sϕt+s=ϕt◦ϕs
U E
CrU CrU E
f
y0=f(y)f(Ut, ϕt)ϕt
U−tUtf
RU f
ϕt∈Diffr(U) = {Crdiffeo U →U}.
t→ϕt
Rn
Cr:I×Um→Rnt mtU
mt(U)
ftmt(U)ft(y) = d
ds ms(x)|s=tmt(x) = y x =m−1
t(y)
X⊂Rn
d Cr∀x∈X
xRnVRn0Cr
ϕ:U→V
ϕ(X∩U) = Rd∩V .
URnf:U→Rn−dCrr≥1
x∈U f x Txf y ∈Rn−d
f y f−1({y})
URnf:U→Rn−d
Crf y f−1(y)
CrdRnf−1(y) = ∅
f:U→RnCrTxf
f:U⊂Rd→Rn
x CrU0x
f(U0)dRn
f:U⊂Rn→Rn−dC∞f
f:U⊂Rp→RnCr
T f x
U0x f(U0)rang(T xf)
y∈f(U0)f−1(y)dim(ker(Txf))

X⊂Cn
dCn
XCd
X⊂Rnx∈
X X x TxX={γ0(0), γ C1,0∈I→
X, γ(0) = x}
TxXRn
d X =f(U)f f(0) = x TxX=im(Txf)
X=f−1(y)f TxX=ker(Txf)
CrX, Y Rn,Rp
Crr0≤r f :X→Y f Cr0∀x∈X
f Cr0
f:X→Y Crf:X→Y→RpCr
x U x Rng:U→Rp
Crg|X∩U=f|X∩V
X, Y f :X→Y C1Txf:TxX→Tf(x)Y
v∈TxX v =γ0(0)
Txf(v)=(f◦γ)0(0) ∈Tf(x)Y .
E, F Rn
E+F=RnX, Y RnX Y
∀x∈X∩Y x X Y
f:X→RnC1f y ∀x∈f−1(Y)
Im(Txf)Tf(x)(Y)
X, Z C1f:X→Zn
C1f Y f−1(Y)
codimXf−1(Y) = codimZn(Y)
X, Y Z ∀x∈X∩
Y, TxX+TxY=TxZ)X∩Y
X Y
X, P, Z Y ⊂Z
F:X×P→Z Fp:X→Fp(x) = F(x, p)
Fp(x)
Y F −1
p(Y)FpY
dim(X) + dim(Y) = dim(Z)F−1
p(Y)
p
X, Y, P RnY⊂Z
F:X×P→Z C∞F
Y p ∈P FpY
f C20
0T0f= 0 0
TfRn× {0}
0Tf:x→(x, Txf)
X C2f:
X→RC2f

f C20Rn
0
ϕRnϕ(0) = 0 T0ϕ=Id
f(x) = f(0) + q(ϕ(x)) q
X⊂RnC∞
l∈Rn∗l|X
X
X
Crn X X
Uii ϕi:Ui→Vi
Rn∀i, j Ui∩Uj6=∅ϕj◦ϕ−1
i=ϕi(Ui∩Uj)→ϕj(Ui∩Uj)
C1ϕiUi
ϕj◦ϕ−1
i2
(Ui, ϕi) (Wα, ψα)ψα◦ϕ−1
iC1
X
d Cr
f:U⊂Rq→RnCr∈U f
U∃U0U f(U0)
f CrU0f(U0)
nPn(R) = {Rn+1}
M∈Mn+1(R)
MPn(R)
fM=fM0⇔ ∃λ∈R, M0=λM .
GLn+1(R)/R∗In+1 =PGLn+1(R)Pn(R)
fM=IdPn(R)=S(M) = R PGLn(R)
F⊂Rn+1
d+ 1 p(F)⊂Pn(R)
d
H1, H2Rn+1 0/∈H1∪H2
(P|H1)−1◦P|H2C∞
X g 6=e∀x∈X gx 6=x
G
∀k{g∈G, gK ∩K6=∅}

X G
X X/G
X/G
Π : X→X/G
ϕi:Ui→Vi⊂Y
Cr
X G
X Y f :X/G →Y Cr⇔f:X→Y
CrG
γ1, γ2C1]−ε, ε[→X
0γ1(0) γ2(0) = x ϕ :U→V⊂Rd
x∈U(ϕ◦γ1)0(0) = (ϕ◦γ2)0(0)
T X ={γ:] −ε, ε[→X/ '} ,
γ1,'γ20
ϕ:X→Y Cr0T ϕ :T X →T Y, [γ]→
[ϕ◦γ]T ϕ Cr0−1