Theorem of Frobenius

Section 4: Theorem of Frobenius
4 Theorem of Frobenius
Suppose M is a dierential manifold and we are given a subbundle E of T
M
. We
seek to nd conditions under which one can assert that at every m M , there exists a
local coordinate system (U, x) such that
x
1
, . . . ,
x
k
, k = rk(E) generate the subbundle
E at all points of U . Note that this implies that all sections of E over U are of the
form
k
i=1
f
i
x
i
. Hence if we consider two sections of E as vector elds and take their
bracket, the resulting vector eld is also a section of E. The theorem of Frobenius asserts
that this condition is also sucient.
4.1. Denition. A subbundle E of the tangent bundle of a dierential manifold M
is said to be integrable if for any two dierentiable sections X and Y of E, the bracket
[X, Y ] is also a section of E.
4.2. Theorem. If E is an integrable subbundle of T
M
, then at every point of M, there
exists a local coordinate system (U, x) such that E|U is generated (freely) by sections
x
1
, . . . ,
x
k
, k = rk(E).
Proof. We will prove this by induction on the rank of E. We have seen in (3.9) that
the theorem is true for k = 1. Let m M. Notice that if X
1
, . . . , X
k
are vector elds
which freely generate E in a coordinate neighbourhood of m, then all of these vector
elds are nonsingular. In particular, X
k
is of the form
x
1
in a suitable coordinate
system (U, x). Writing out the other X
i
’s in terms of the basis
(
x
i
)
, we see that by
subtracting from each of X
i
, i < k, a multiple of X
k
, one may assume that they are all of
the form
i2
f
i
x
i
. Direct expansion of [X
i
, X
j
], i j < k, shows that the subbundle
of E generated by X
i
, i < k is also integrable. Our induction assumption then implies
that there is a coordinate system (V, y) at m such that X
i
=
y
i
for all i < k. In this
coordinate system, suppose X
k
=
f
i
y
i
. Again by subtracting from X
k
the linear
combination
i<k
f
i
y
i
of X
i
, i < k, we may assume that X
k
=
ik
f
i
y
i
. Moreover,
since X
k
is nonsingular, at least one of the f
i
’s, say f
k
, is invertible. We may replace
X
k
by X
k
/f
k
. Then we have X
i
=
y
i
for all i < k and X
k
=
y
k
+
i>k
g
i
y
i
. Now
[X
j
, X
k
] =
i>k
g
i
y
j
y
i
on the one hand, and is a linear combination of X
i
, i k on
the other. This implies that
g
i
y
j
are all zero. In other words, g
i
are independent of
y
j
, j < k. Let us now take the coordinate neighbourhood in the form V
1
× V
2
with
V
1
(resp. V
2
) a domain in R
k1
(resp. R
nk+1
) with coordinates y
1
, . . . , y
k1
(resp.
y
i
, i k). Since X
k
can be regarded as a nonsingular vector eld in V
2
, we can nd a
coordinate system z
1
, . . . , z
nk+1
in which X
k
has the expression
z
1
. Now it is clear
that (y
i
, 1 i k 1, z
j
, 1 j n k + 1) is a coordinate system with the required
property.
4.3. Denition. An immersed manifold φ : N M is said to be integral for a
53
Chapter II: Differential Operators
subbundle E if the dierential of φ at any point p N maps the tangent space T
p
(N)
isomorphically onto the bre E
φ(p)
.
In view of Theorem 4.2, there do exist integral submanifolds for an integrable sub-
bundle. In fact, take a coordinate system (U, x) as in Theorem 4.2 (where for convenience
of notation we will assume that U is an open cube in R
n
in the coordinate system). Then
the closed submanifolds S
(a)
= {(x) : x
i
= a
i
for all i > k}, are obviously all integral
manifolds. An integral manifold obtained in this way is called a slice.
Suppose φ : V U is any connected immersed manifold which is integral for E.
Then the functions x
i
φ satisfy
φ
i
x
j
= 0 for all i > k and j k. By our assumption,
this implies that vφ
i
= 0 for all i > k and all tangent vectors v at any point of V .
Hence φ
i
is a constant on V . The manifold V is therefore contained in the slice S
(a)
,
and indeed as an open submanifold.
4.4. Remark. If φ : M N is an integral manifold, then for every p N , there
exists a neighbourhood U such that φ(M )U is a countable union of closed submanifolds
of U . We may of course also assume that these are connected components of φ(M ) U.
4.5. Exercise. Explain in the above light which of the examples of immersed sub-
manifolds given in [Ch. 1, 3.17] are integral curves and which not, for a suitable line
subbundle of the tangent bundle.
We will now show that immersed integral manifolds do not admit pathologies of the
kind we pointed out in [Ch. 1, 3.17].
4.6. Proposition. If φ : N M is any integral manifold and L is any other manifold
with a map f : L N, then f is dierentiable if and only if φ f is.
Proof. If f is dierentiable, the composite φ f is of course dierentiable. In proving
the converse, the key point is that the dierentiability of φ f implies the continuity
of f. Let l L and (U, x) be an open cube containing φ f(l) as in Theorem 4.2.
The open submanifold (φ)
1
(U) of N is a countable union of open connected manifolds,
each of which is an open submanifold of a slice. The map φ f maps a connected open
neighbourhood W of l into the union of these slices. But the image in R
nk
is countable
and connected and hence consists of a single point. In other words, the map φ f maps
W into a locally closed submanifold of M . Hence it is dierentiable as a map into the
submanifold as well.
4.7. Corollary. If φ : N M and φ
: N
M are two connected integral manifolds
for E with the same image, then they are dieomorphic.
Proof. Indeed the natural maps N N
and N
N are both dierentiable, by
Proposition 4.6.
54
Section 4: Theorem of Frobenius
We can now take the set of all (connected) integral manifolds for a given integrable
subbundle E (identifying them with their images) and partially order them by inclusion.
Clearly it is an inductive family and therefore there exists a maximal element. These
are called maximal integral manifolds.
We will now give an application of the Frobenius theorem to Lie groups.
4.8. Corollary. Let G be a connected Lie group and h a Lie subalgebra of the Lie
algebra g of G. Then there is a Lie subgroup H of G with h as its Lie algebra.
Proof. Consider the subbundle E which is left G invariant such that E
1
= the subspace
h of T
1
= g. Its sections are generated over A by a basis (X
i
) of the vector space h. In
other words, all sections are of the form
f
i
X
i
. If X =
f
i
X
i
and Y =
g
i
X
i
, then
[X, Y ] =
f
i
(X
i
g
j
)X
j
g
i
(X
i
f
j
)X
j
+
f
i
g
j
[X
i
, X
j
] is again a section of E. Hence
E is integrable. Let H be the maximal integral submanifold for E containing 1. For any
h H, the left translation by h
1
of H gives another maximal integral submanifold for
the same integrable subbundle, since E is invariant under left translations. But since
h H, this translate contains 1 and so coincides with H. Hence h
1
H and H is
closed under multiplication. In order to show that H is a Lie group, we have to verify
for example that the map H × H H given by (h
1
, h
2
) 7→ h
1
h
1
2
is dierentiable. By
Proposition 4.6, it is enough to check that this map, considered as one from H ×H into
G, is dierentiable. But this latter map is the composite of the inclusion of H × H in
G ×G and the corresponding group multiplication map of G ×G into G. This completes
the proof.
4.9. Remarks.
1) Even in the case of a Lie subgroup H the topology of H may not coincide with
that of the image, as the illustration in [Ch. 1, 3.17] shows. However we have the
following comforting situation in the case of Lie groups. Since H induces injection
on the Lie algebra, it is an integral submanifold of G for the left invariant subbundle
of the tangent bundle of G dened by h. Hence the subgroup H can be provided
with the structure of an immersed manifold.
2) A connected Lie subgroup is a locally closed manifold only if it is actually a
closed submanifold. For its closure is a connected subgroup in which it is open.
But an open subgroup is necessarily closed. In this case, let p be a subspace of
g, supplementary to h. The exponential image of this space in the exponential
neighbourhood intersects H only at 1. The coset space G/H is a Hausdor topo-
logical space with a countable base for open sets. Besides there is a neighbourhood
of the trivial coset which can be provided with a dierentiable structure, via the
exponential map. From this we easily conclude that G/H is a dierential manifold
and that the natural map G G/H admits a dierentiable section.
55
Chapter II: Differential Operators
3) One can show that if H is a closed subgroup of G then it is automatically a Lie
subgroup, and therefore the above considerations do apply.
4) It can also be proved that an arcwise connected subgroup of a Lie group is a Lie
subgroup.
Suppose that H and G are connected Lie groups with h, g as their Lie algebras.
Given any Lie algebra homomorphism t of h into g, consider the graph of t, namely the
Lie subalgebra of h × g dened by the set of elements (X, tX), X h. Let
˜
H be the
corresponding connected Lie subgroup of H × G. The projection homomorphism of
˜
H
into H induces an isomorphism of Lie algebras. Hence there exists a discrete normal
subgroup N such that
˜
H/N is mapped isomorphically on H. Thus although the map t
may not come from a homomorphism of H into G, there is an étale covering of H from
which there is a homomorphism giving rise to t.
4.10. Remark. We have carried out the correspondence between Lie groups and Lie
algebras except for one particular. It is also true that every Lie algebra is actually the
Lie algebra of a Lie group. This would follow for example, if we can show that there
is an injective homomorphism of the Lie algebra into gl(n, C), in view of Corollary 4.8.
This latter assertion is known as Ado’s theorem, and we do not prove nor use it in this
book.
56