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5.3. FROBENIUS THEOREM
123
Thus, span
{
v
i
}
= Δ
.
7. We have
d
α
μ
=
X
μ<ν
C
μλν
α
λ
∧
α
ν
+
X
i
,ν
A
μλ
i
β
i
∧
α
ν
+
X
i
<
j
B
μ
i j
β
i
∧
β
j
=
X
ν
γ
μν
∧
α
ν
+
X
i
<
j
B
μ
i j
β
i
∧
β
j
.
8. Thus,
B
μ
i j
=
(
d
α
μ
)(
v
i
,
v
j
)
=
0
.
9. (3)
=
⇒
(4). Suppose that
d
α
μ
∧
ω
=
0.
10. Then we have
0
=
d
α
μ
∧
ω
=
X
ν
γ
μν
∧
α
ν
∧
ω
+
X
i
<
j
B
μ
i j
β
i
∧
β
j
∧
ω .
=
X
i
<
j
B
μ
i j
β
i
∧
β
j
∧
α
1
∧ ··· ∧
α
n

k
.
11. Thus,
B
μ
i j
=
0.
±
•
Remarks.
•
A
k
dimensional distribution
Δ
can be described locally by either
k
linearly
independent vector ﬁelds that span
Δ
or by (
n

k
) linearly independent
1forms whose common null space is
Δ
.
•
If
d
α
μ
=
∑
n

k
ν
=
1
γ
μν
∧
α
ν
for some 1forms
γ
μν
, then we write
d
α
μ
=
0 (mod
α
)
.
5.3.2
Frobenius Theorem
•
Deﬁnition 5.3.4
Let M be an ndimensional manifold, W be a k
dimensional manifold and F
:
W
→
M be a smooth map.
Then F is an
immersion
if for each x
∈
W the di
ﬀ
erential F
*
:
T
x
W
→
T
F
(
x
)
M is injective, that is,
Ker
F
*
=
0
.
The image F
(
W
)
of the manifold W is called an
immersed submani
fold
.
di
ﬀ
geom.tex; April 12, 2006; 17:59; p. 123
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CHAPTER 5. LIE DERIVATIVE
•
Let
M
be an
n
dimensional manifold.
•
Let
Δ
be a
k
dimensional distribution on
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.
 Spring '10
 Wong
 Geometry

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