(Uj, ψj)MUj
M j ϕ(Uj)
N j
ϕ(C∩ Uj)
M=URmN=Rn
m m = 0
C≡C0ϕ
Ck=½x∈ U :∂αϕ(x)
∂xα= 0,∀α∈Nm|α| ≤ k¾, k ∈N∗
C≡C0⊃C1⊃C2⊃ ··· ⊃ Ck⊃Ck+1 ⊃ ···
ϕ(Ck\Ck+1)
(i)ϕ(C0\C1)ξ∈C0\C1
ϕ1, ..., ϕnϕ ξ /∈C1
1ϕ ξ
∂ϕ1
∂x1(ξ)
f:U −→ Rm,(x1, ..., xm)7−→ (ϕ1(x1, ..., xm), x2, ..., xm),
m ξ
VξUf
V W ≡ f(V)f(ξ)RmC0\C1
Vϕ
g≡ϕ◦f−1:W −→ Rn,
U W gWRn
g ϕ V
C0g f(V ∩C)g(C0) =
ϕ(V ∩C)g prRk
Rk
prRn.ϕ =prRm(t, x2, ..., xm)∈ W
g g(t, x2, ..., xm)t×Rn−1
t
gt:¡t×Rm−1¢∩ W −→ t×Rn−1,
g
det µ∂gi
∂xj¶= det µ∂(gt)i
∂xj¶,