216 - Etudes métriques de courbes. Exemples. - IMJ-PRG

Rn
Rn(I, f)I
f I Rnf(I)
supp(I, f)
RnC1
k C1
C2Ckk
RnC1f:I→R
IS1
t
f(t)
(I, f) (J, g)
CkCk
Ckθ:I→J g =f◦θ Ck
Ck
Ck
θ
(I, f) (J, g)
Ckθ:I→J g =f◦θ
t7→ eit ]−π, π]R
i∈N(I, f)t0∈I
V ect(f0(t0), . . . , f(i)(t0))
i
t0
t0
t0
t0Rf0(t0)
V ect(f0(t0), f00(t0))
p q
p
p q
p q
p q

p q
n= 3
t0= 0 x(t)y(t)z(t)
f(t)−f(t0)f0(t0)f00(t0)f000(t0)
x(t)∼t y(t)∼t2
2z(t)∼t3 6
γ= ([a, b], f)S
σ= (t0=a<t1<· · · < tm=b) [a, b]
γR+supσ∈S Pm
i=1 d(f(ti−1), f(t))
l(γ)
t∈[a, b]ϕ(t) = l(([a, t], f|[a,t]))
ϕ
f
t, t0ϕ(t)−ϕ(t0) = t−t0(I, f)
(J, g)f(t) = g(±t+T)
([a, b], f)C1
Rb
akf0(t)kdt
2π
π
x(t) = t−sin t, y(t) =
1−cos t
S1R
C1γ
C1R2R2\γ
KR2
P dx +Qdy C1
K
Z∂K+
(P dx +Qdy) = ZZK∂Q
∂x −∂P ∂y dxdy
K
A=RRKdxdy
A=Z∂K+ xdy =−Z∂K+ ydx =1
2Z∂K+
xdy −ydx =1
2Z∂K+
r2dθ
r2=a2cos 2θ θ ∈[−π/4, π/4]
a > 0
A=1
2Zπ/4
−π/4
a2cos 2θ dθ =a2
2
γ: [0,1] →C
L
S L2≥4πS γ
ΓCkk≥2
γ= (I, f)
n= 3
Γs
c(s) = kf00(s)kc(s)6= 0
Γs R(s)p(s) = f(s) + R(s)f0(s)
Γs p(s)R(s)
Γs

s c(s)6= 0
s
f0(t)f00(t)
c=kf0∧f00k
C2
f
(J, g)
Γg=f◦ϕ
c(t) = kg0(t)∧g00(t)k
kg0(t)k3
g(θ) = aur+bθuzc(θ) = a
a2+b2
b= 0
ρ(θ) = 1 + 2 cos θ c(θ) = 9+6 cos θ
(5+4 cos θ)3/2
n= 2 RnC
γ s c(s)
f00(s) = c(s)if0(s)
c= [f0, f00]
Γ
C3Γ
(J, g) Γ
c(t) = [g0(t), g00(t)]
kg0(t)k3
Γ
x(t) = acos t
y(t) = bsin t
(ζ(t) = a2−b2
acos3t
η(t)−a2−b2
bsin3t
s τ(s) = f0(s)ν(s) = f00 (s)
kf00 (s)kβ(s) = τ(s)∧ν(s)
Γs
R3(f(s), τ(s), ν(s), β(s))
Γs

C3β0(s)
ν(s)s γ(s)β0(s) =
γ(s)ν(s)
C3
τ0=cν, ν0=−cτ −γβ, β0=γν
(J, g)
Γg=f◦ϕ
τ=g0
kg0kβ=g0∧g00 kg0∧g00kν=β∧τ γ =−[g0, g00, g000] kg0∧g00k2
τ=
a
√a2+b2uθ+b √a2+b2uzν=−urβ=−b √a2+b2uθ+a √a2+b2uzγ=−b
a2+b2
IRϕ∈C1(I, R∗
+)ψ∈C0(I, R)
(s0, M0, U0, V0)∈I×R3×(S2)2U0
V0C3
(I, f)ϕ
ψ f(s0) = M0f0(s0) = U0f00(s0) = kf00(s0)kV0
t7→ (t, e−1/t,0) t > 0
t7→ (t, 0, e1/t)t < 0
07→ (0,0,0) 
t7→ (t, e−1/t,0) t > 0
t7→ (t, e1/t,0) t > 0
07→ (0,0,0)
C1