Rapport de stage L3

H2
H2
γ
L(γ) = ´kγ0(t)kγ(t)dt
γ0(t)γ(t)
H=(x1, x2, x3)∈R3|x2
1+x2 2−x2 3=−1, x3>0

C∞R3
k(x1, x2, x3)k(x,y,z)=x2
1+x2 2−x2 3
f:(R3→R
(x, y, z)7−→ x2+y2−z2+ 1
Df(x,y,z)= (2x, 2y, −2z)
T(x,y,z)H=KerDf(x,y,z)= (x, y, z)⊥
p:(H−→ R2
(x1, x2, x3)7−→ (x1, x2)
< Dp(u)|Dp(v)>p(x)=< u|v >x
x2
3<(u1, u2)|(v1, v2)>(x1,x2)=x2 3(u1v1+u2v2−u3v3)
x2
1+x2 2−x2 3=−1x3u3=x1u1+x2u2
<(u1, u2)|(v1, v2)>(x1,x2)=u1v1+u2v2+(x1u2−x2u1)(x1v2−x2v1)
1+x2
1+x2 2
ψ(H−→ D
(x, y, z)7−→ (x
z+1 ,y z+1 )
ψR2
φ:(D−→ H
(x, y)7−→ (2x
1−x2−y2,2y 1−x2−y2,1+x2+y2 1−x2−y2)
φ
Jφ(x, y) = 1
(1−x2−y2)2
2(1 + x2−y2) 4xy
4xy 2(1 −x2+y2)
4x4y

φ
< u|v >x,y=< Jφ(x, y).u|Jφ(x, y).v >φ(x,y)
< u|v >x,y=4(u1v1+u2v2)
(1−x2−y2)2
C:z7−→ C(z) (C(z) + i)(z−i)=2
C|P
D
C−1(z) = 1+iz
z+i
(C−1)0(z) = −2
(z+i)2
< u|v >z=<(C−1)0(z).u|(C−1)0(z).v >C−1(z)
< u|v >z= 16 <u|v>euc
(|z+i|2−|1+iz|2)
(|z+i|2− |1 + iz|2) = 4Im(z)
< u|v >z=<u|v>euc
Im(z)2
H2
cos(θ) = <u|v>z
kukzkvkz
<|>euc

f:(D−→ D
z7−→ λz−z0
zz0−1|λ|= 1 |z0|<1
f(D)⊆D
∀z1∈D f(1
λz1−z0
1−1
λz1z0) = z1
f0(z)6= 0 ∀z∈D
z7−→ f(z−z0
zz0−1)z0=f(0)
z7−→ αz
z7−→ az+b
bz+aa2−b2= 1
z7−→ eiθ z−z0
zz0−1
z7−→ az+b
bz+aa=e
i√1−z2
0
iθ/2b=e
i√1−z2
0
−iθ/2z0
z7−→ az+b
cz+dad −bc = 1
C−1ofoC az+b
bz+a