TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 180, June 1973
ORBITS OF FAMILIES OF VECTOR FIELDS
AND INTEGRABILITY OF DISTRIBUTIONS
BY
HECTOR J. SUSSMANN
ABSTRACT.
Let D be an arbitrary
set of C
vector
fields
on the C manifold M. It is shown that the orbits ol D are C
submanifolds
of M, and that,
moreover,
they are the maximal
integral
submanifolds
of a certain
C
distribution P„.
(In general,
the dimension
of PrJm)
will not be the same for all m cM.)
The second
main result
gives necessary
and sufficient
conditions
for a distribution to be integrable.
These
two results
imply as easy corollaries
the theorem
of Chow about the points
attainable
by broken integral
curves of a family of vector fields,
and all the known results
about integrability
of distributions
(i.e. the
classical
theorem
of Frobenius
for the case of constant
dimension
and the more
recent
work of Hermann,
Nagano,
Lobry and Matsuda).
Hermann
and Lobry studied orbits in connection
with their work on the accessibility
problem
in control
theory.
Their method
was to apply Chow's
theorem
to the maximal
integral
submanifolds
of the smallest
distribution
A such that every vector
field X in the
Lie algebra
generated
by D belongs
to A (i.e.
X(zzz) e A(zzz) for every
m eM).
Their work therefore
requires
the additional
assumption
that A be integrable.
Here the opposite
approach
is taken.
The orbits
are studied
directly,
and the
integrability
of A is not assumed
in proving
the first main result.
It turns out
that A is integrable
if and only if A = P„,
and this fact makes
it possible
to
derive
a characterization
of integrability
and Chow's
theorem.
Therefore,
the
approach
presented
here generalizes
and unifies
the work of the authors
quoted
above.
1. Introduction.
Let D be a set of C°° vector fields
We are interested
local
generators
sociate
with
signs
a linear
subspace
precise
maximal
C
subspace
in §3).
passes
integral
Received
D-orbits
that
Pp
of the tangent
This
by the editors
(i.e.
let
has
integral
(with
of the action
a natural
to M at
the property
of
the orbits
P„
to be
that
C
through
and that,
that
how to asm £ M as-
ttz; the dimension
is supposed
of G.
topology
to each
which
of
infinitesimal
we show
manifold
are precisely
whose
are the orbits
D-orbits
M.
G be the "group"
groups
of M. Moreover,
space
distribution
C°° manifold
a mapping
ttz, and the mapping
a maximal
manifolds
the
submanifolds
distribution
may vary with
of M there
The
4.1) states
in §2) are
D a C
Precisely,
by the one-parameter
of D.
(Theorem
that we define
the zO-orbits.
generated
are the elements
Our main result
made
in studying
diffeomorphisms
on the
of this
in a sense
every
moreover,
point
these
of D.
June 13, 1972.
AMS(MOS)subject classifications
(1970). Primary 34C35, 34C40, 58A30, 49E15,
'
Copyright © 1973, American Mathematical Society
171
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172
H. J. SUSSMANN
The orbits
erature
of an arbitrary
because
Sussmann
of their
by Chow
Let
denote
D
under
18]).
[2], and on the theory
m £ M, then the orbits
the notation
theorem
bits.
ting,
for each
applied
of integral
theorem
fields
says
it is still
integral
hull
manifolds
studied
[4], Lobry
submanifolds
D and is closed
maximal
rank
components
D a distribution
J ÍD),
of D im).
Chow's
Then
for each
of M (here
if the assumption
of J(D),
[5jj
on a
of distributions.
contains
if D im) has
Even
in the lit-
so far relies
of Chow's
to get a good description
with
ttz £ M, A(D)(ttz) = linear
to the maximal
used
the connected
possible
one associates
been
(cf. Hermann
which
that
of D are precisely
this
have
has been
DAm) = iX(77z) : X £ D \).
is not satisfied,
To achieve
that
set of vector
Chow's
fields
in Control Theory
The method
the smallest
Lie brackets.
we use
D of vector
importance
and Jurdjevic
theorem
set
of the or-
defined
and it follows
by let-
theorem
that
is
these
mani-
folds are the orbits of D.
The method
the maximal
integral
ple examples
folds,
in the preceding
manifolds
(cf. §3).
even
suggests
outlined
though
fact
speaking,
the orbits
mal integral
on Chow's
general,
to those
every
point
characterization
An obvious
necessary
This
manifolds,
Roughly
are given
as maxi-
why the method
Pp.
is simply
The
Chow's
"right"
theorem
based
that,
in
distribu-
can be applied
Pp - J (/?)•
obtain
a complete
integral
a maximal
integral
characterization
manifolds
of
property
manifold
and it implies
We shall
condition
is that it be involutive
the dimension
the classical
A(ttz) varies
[7] proved
theorem.
of A).
(i.e.
This
all the known results
now indicate
how this
result
relates
work.
manifolds
Hermann
of distributions.
with
by sim-
submani-
assertion.
they
cases
the maximal
4.2,
are still
The reason
in which
when
passes
in Theorem
this
the pathological
we shall
have
of M there
is given
integrability
to previous
A that
Pp.
can be seen
from the integral
Moreover,
not coincide
cases
of our work,
C°° distributions
through
distribution
to handle
drawback:
of Chow's
arises
4.1 substantiates
and the situations
precisely
As a by-product
This
the orbits
by means
behaved.
J (/?) does
Pp,
an obvious
not exist.
be proved
well
of a C
is unable
has
examples
when it occurs,
Our Theorem
the distribution
correspond
analytic
cannot
are always
manifolds
theorem
tion to look at is
about
in these
that the "pathology",
and not from the orbits.
those
of J (D) need
However,
this
paragraph
of A(ttz) is constant,
theorem
(i.e.
when
[4] stated
of Frobenius,
A has
various
that integrability
distribution.
Lobry
for a C°° distribution
(the definition
then
this
condition
cf. Chevalley
"singularities"),
conditions
follows
Ll]).
extra
that would
if M is a real
[5] introduced
A to have
is given
in §3).
is also
assumptions
analytic
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sufficient
When the dimension
guarantee
a condition
integral
If, in addition,
(that
is
of
are needed.
integrability.
manifold
(this
and
A be "locally
Nagano
A is an
of
173
ORBITS OF FAMILIES OF VECTOR FIELDS
finite
type")
which
automatically
Nagano's
result.
vergence
condition".
sists
claim,
that
we have
of our work,
as the fundamental
we wish
to make this
of our approach,
feomorphisms,
and we introduce
explains
its geometric
proofs
of Theorems
derive
Chow's
marks
about
Nagano's
(a)
case
two remarks
the analogy
of local
theorem.
the basic
groups
contains
of local
the definition
is meant
of Theorem
in particular,
Lobry,
The
In §7 we
4.2 with
we derive
In §9 we make
following
of the
at the end of the section
respectively.
and,
dif-
by a C°° distri-
our two main theorems.
and Matsuda.
the
and
of the advantages
in §5 and §6,
and indicate,
the
a few re-
how to derive
on terminology.
to speak
rather
in the rest
diffeomorphisms
to
acting
on a manifold.
These
in the algebraic
sense,
For a similar
reason,
we use the
The reader
who wishes
[4], Lobry
for "group"
[5]).
and/ or "leaf"
article
by AD*
of vector
fields
and orbits.
according
to the general
they
for "orbit".
which we have denoted by J (D) (following
of this
unless
[8]) will be
notational
of "§3.
2. Families
are supposed
(Hermann
"pseudogroup"
of local
of a Lie group
diffeomorphisms.
"leaf"
(b) The distribution
"groups"
are not groups
defined
than
to do so may substitute
conventions
about
with the case
diffeomorphisms
of everywhere
word "orbit"
denoted
Lobry
is that
problems,
the connection
of distributions
this
proofs
is only acquainted
fields,
A remark
In §4 we state
Hermann,
for this
Chow's
what
(ra-
one obtains
in §2 we introduce
section
Pp.
at orbits
To substantiate
which
of vector
this
In V8 we discuss
We have chosen
emphasize
consist
only what
and included
reason
imply
In §3 we define
4.1 and 4.2 are given
the analytic
con-
result.
Finally,
groups
families
In particular,
on integrability
by looking
As an illustration
is as follows:
the distribution
of Frobenius,
from a "con-
and Lobry
of controllability
by an audience
geometry.)
meaning.
theorem.
known results
theorems
readable
of the orbits.
bution,
is
Its proof
to be studied,
(Another
concerning
and orbits.
follows
many known results.
are to the study
of the paper
topology
that,
objects
we show in §7 how our results
and notations
"natural"
we submit
that are not new.
article
condition
a new proof of
results.
of Hermann
self-contained,
of differential
The organization
also
all these
practically
of our results
with the rudiments
definitions
and unifies
paper
of results
main applications
this
integrability.
manifolds)
clarifies
since
work provides
integrability
4.2 contains
from the conditions
to obtain
made this
(by our methods)
that
Lobry's
by-product
integral
an approach
Moreover,
case,
[6] showed
of isolating
As a second
integrability.
Our Theorem
necessary
ther than
implies
in the analytic
Matsuda
essentially
is strictly
also
satisfied
to be of class
Throughout
C°° and paracompact.
this
paper,
If M is a manifold,
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all manifolds
a submani-
174
H. J. SUSSMANN
fold of M is a manifold
map from
S into
particular,
S such
M is a C
the inclusion
pological
subspace
We shall
use
field
to denote
field
and
injective
but it is not required
the tangent
of M . In particular,
denote
space
that
to the manifold
(in
S be a to-
the meaning
to the submanifold
the set
of all
which
Y, and which
is defined
is given
M at the point
can be identified
in a natural
of expressions
such
way
as "the
S" is clear.
C°° vector
fields
defined
M. If X and Y belong to V(M), then the Lz'e bracket
vector
the inclusion
is everywhere
of M, and if s £ S, S
X is tangent
VÍM) will
of M and that
differential
map is continuous,
M
a subspace
vector
S is a subset
of M).
77Z. If S is a submanifold
with
that
map whose
on the intersection
on open
subsets
of
[X, Y] of X and Y is a
of the domains
of definition
of X
by the formula
[X, Y]/= X(Yf)- Y(Xf)
for / a C
real-valued
function
"provided
the domains
tor field
to be an element
be introduced
later,
of X and
such
Z in the domain
By well-known
as local
y of X such
which
contains
the interval
(as well
ping
as
curve
write
subset
t, X
fi'.
clear that
/).
ttz. To indicate
X im) instead
is a diffeomorphism
to
in an open
refer
It is well
subset
fi(X)
from an open subset
for
Xiyit)).
an integral
interval
/
in addition,
the dependence
of yit).
t is
of an integral
We shall
that,
that
curve
to this
which
curve
on X and
known
that
of R x M into
ttz
the mapM.
fi of M onto an open
The set fi (which may be empty) will be denoted by fi (X). It is then
fi
We shall
is precisely
need
If A is an arbitrary
fi
(X).
some notations
set,
let
for composites
Am denote
of several
maps
the set of all (ordered)
(Zj, • ■■ , t ), then £Tim)
will denote
12
that
use
X" (m).--)).
n
£fTim) will be defined
of Rm x M. We shall
, Xm) and T =
the point
X1 (X2(...
It is clear
of the form X
zzz-tuples of ele-
ments of A. Let £ £ ViM)m and T £ Rm, m £ M. If f = (X1, ...
subset
such
exists
if we require,
contains
it, m) —» X im) is a C°° map from an open
For each
vec-
concepts
to y at
of X there
is defined
y is unique
properly
of X through
z) we shall
as
the empty
Z —»yit)
if y is not the restriction
which
such
to other
vector
ttz in the domain
Moreover,
(i.e.
on an interval
as the integral
of y, the tangent
y(0) = 77z, and which
/ be maximal
is defined
declare
applies
of X is a C°° curve
for every
that
remark
for remarks
diffeomorphisms.)
curve
the origin.
the need
we shall
A similar
of definition
theorems,
curve
Y intersect",
of V(M).
If X € VÍM), an integral
every
on M. (To avoid
for all
fi(<f ) to denote
(T, ttz) which
this
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set,
and
belong
to an open
fiT(rf ) to denote
ORBITS OF FAMILIES OF VECTOR FIELDS
the set
of all
ttz such
that
(T, ttz) £ fi(<f)
(i.e.
the set
of all
175
m for which
çJm)
is defined).
A local
diffeomorphism
of M onto an open
subset
then the composite
range
S,(fi2
morphism
on M is a C
fi.
8X82
O fi,).
If 8 '. : fi. —> fi.
is a local
The
with domain
inverse
fi,
diffeomorphism
of
fij.
a local
called
of local
under
diffeomorphisms
compositions
diffeomorphism
the group
let
Gp to denote
generated
diffeomorphisms
which
this
group,
by D.
quence
contains
It is clear
that
call
is everywhere
Let
say that
that
a subset
of the elements
notation,
of the
integer
the operations
(£T)_1
M. Similarly,
clearly
modulo
this
D, or D-orbits.
77z2. Equivalently,
y:
defines
relation
defined,
Two points
integer
are called
ttZj and
rz, there
ttZj and
ttz2 belong
[a, b] —> M such
that
of Gp
are given
of some
if there
by:
T £ R"
orbit
of the doG
g £ G.
on M. We
is a g £ G such
on M. The equiv-
of G (or G-orbits).
to the same
to the same
y (a) = ttz,, yib)
Xp
the se-
diffeomorphisms
will be referred
<f £ D" and
T £ R".
we use
if the union
relation
the orbits
ttz2 belong
exist
use
the mappings
diffeomorphisms
an equivalence
the GD-orbits
of lo-
We shall
A will denote
of local
ttz of M are G-equivalent
ttz and
This
by Gx.
diffeomorphisms
sequences,
to the domain
group of local
is
group
n, ^ £ D" and
defined
a group
m £ M belongs
is
= £_ f..
two elements
for some positive
of local
of GD are precisely
defined
classes
a smallest
GY for X £ D.
D of V(M) is everywhere
if every
D C V(M) is everywhere
a curve
exists
it the group
and
X
by X, and is denoted
There
G be an everywhere
g(7Tz,)= ttz,.
alence
diffeomorphisms
local
p = (px, • ■• , p.) are finite
of D is
defined
the mapping
for some positive
With this
which
of all such
(Xx, ■■■ , X , px, • ■■ , pA,. Also,
(X , • • ■ , Aj).
We say that
diffeomorphisms
If X e VÍM) then
the elements
where,
, X ) and
the sequence
G of local
the union
^j.rjj.1 = (¿;t))tt'
mains
diffeo-
=S2-1S71
generated
of VÍM).
and we shall
are of the form çT
If X = (A.j, ...
is a set
Z. The set
D be a subset
cal diffeomorphisms
to denote
and
laws
(S^)"1
and inverses.
for every
of local
More generally,
which
8~ (fi.)
by 87 , and is a local
The formal
fi
valid.
A group
is closed
subset
diffeomorphisms,
with domain
<5. is denoted
(S1S2)z53= z51(/52S3) and
are clearly
(z = 1, 2) are local
diffeomorphism,
and range
from an open
If
to as the orbits
orbit
if and only
such
if and only
= m2, and which
that
if,
¿jT(mA
if there
has
of
exists
the follow-
ing property:
(PI)
fields
There
exist
numbers
Z. such
X1 £ D (i = 1, • . ■ , r) such
that,
that
a = Iq<
for each
tx < • ■• < t = b and vector
z, the restriction
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=
of y to
176
lt.
H. J. SUSSMANN
,, tx1 is an integral
A curve
curve
y which
of X' or of —X1.
satisfies
condition
(PI) will
be called
a piecewise
integral
curve of D.
The orbits
Pf
denote
the map
be the orbit
pings
pp
(for all
of D can be given
M. Since
i : Sm —* Sm
and
p¿
fi¿r
Odr^, / . Since
rÇV,m
to every
A set
Aim)
for every
A distribution
is the linear
fields
hull
into
ç,
defined
subset
m in the domain
long to A.
It is clear
If G is a group
to be G-invariant
ttz belongs
then
dg~
same
for all points
We let
of local
ttz which
D¿ denote
A(ttz) into
A(g(7rz)) into
A(ttz).
belong
A2 are distributions,
X belongs
is a smallest
then there
The space
v £ Aim)
or v = dgiw)
where
and
distribution
A (ttz) is the linear
g £ G and,
Oc
rÇ ,rn
: fit Ç ,m —>Sm
A which
space
as-
M .
ttz £ M,
to the given
set.
is spanned
If
by D.
a C°° distribution.
X £ VÍM) which
be-
if and only if A is spanned
by
the distribution
A is said
ttz £ M and every
g such
and if g, m are as above,
the dimension
of A(ttz) is the
G-orbit.
we say that
phisms,
r)T (m ) = m.
of T—>TTQ
A if X(ttz) £ Aim) for
If A is G-invariant
ttz. If A is a distribution
map
which is of the form Aß
for every
In particular,
that
A which
on M, then
to a given
the inclusion
that
the set of all
Aigim))
C A2(t7î) for every
variant.
defined
that
A if, for every
D of VÍM) is called
diffeomorphisms
of g.
the topology
M is a mapping
the distribution
A is a C°° distribution
to the domain
maps
and
that
if dg maps
that
If A,
of X.
of
follows.
A vector field X £ VÍM) belongs to the distribution
every
of
subspace
on the
Sm ,' it follows
our conclusion
is a distribution
pc
M is continuous.
is the composite
will be denoted by AD. A distribution
everywhere
all the
not depend
q, TQ be such
X(?rz), where
there
S
S is connected.
A(ttz) of the tangent
to span
of the vectors
defined,
that
on a manifold
subspace
is said
makes
S into
to prove
Let
Now let
of all the map-
a topological
S with
(m ), which
is continuous
D C VÍM) is everywhere
for some
ttz, ttz.
let
of S as a subspace
on S does
Sm denote
T —> ^j-rjj-
m £ M, a linear
it follows
It is sufficient
is true for all
of vector
This distribution
.
this
3. Distributions.
signs,
pe
p.,
rST),77!
Since
which
map from
so defined
ttz £ S, let
of the maps
is continuous
is continuous.
are connected
S ,777
is the mapping
topology
the topology
the inclusion
ç £ Dn,
be its domain.
of the images
S will not be, in general,
For each
by means
Then
that
If ttz £ M and
(Ç Rn)
Since
that the topology
of ttz £ S.
above
it follows
all the sets
fizr
S is the union
continuous.
S is Hausdorff.
We now verify
topology.
S by the strongest
ç £ D")
property,
In particular,
and let
ttz. Then
. We topologize
n and all
M has this
choice
T —» £,Tim)
of D through
a natural
A
A,
is contained
G is a group
which
contains
hull of all the vectors
for some
in A2 if Aj(t?z)
of local
A and is G-in-
v £ V
m' £ M, ttz = gimA
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diffeomor-
such
and
that
w £ AW).
ORBITS OF FAMILIES OF VECTOR FIELDS
If A is spanned
of all vector
that belong
by D C VÍM), then
fields
which
to G.
Let
a distribution
D-invariant
Ap,
i.e.
which
by the union
of D under
is G^-invariant
which
defined
subset
remarks
that
of ttz. If 5 is a D-orbit,
local
diffeomorphisms
A
We'are
which
is also
by A
interested
is D-invariant
Clearly,
C°°.
to be D-invariant.
A is denoted
of V(M).
distribution
of D and the set
then
is said
contains
will be denoted by Pp.
from the preceding
the D-orbit
of elements
distribution
the smallest
This distribution
It follows
is spanned
if A is a C°° distribution,
D be an everywhere
distribution
Ap.
are images
In particular,
If D C VÍM),
The smallest
A
177
.
in the
and contains
Pp is a C°° distribution.
the dimension
of Pd(t7z) depends
and if m £ S, then
the dimension
only on
of Ppim)
is
called the rank of the orbit S.
A set D C VÍM) is said
follows
that
smallest
to be involutive
[X, Y] belongs
involutive
to D.
subset
if, whenever
X £ D and
If D C V(Al) is arbitrary,
of V(M) which
contains
D.
then
This
Y £D,
there
it
exists
a
set will be denoted
by D*.
A C°° distribution
shown
Since
below
every
that,
A is involutive
if the set
if D C V(M) is everywhere
X £ D belongs
to Pp, we have
(3-D
It is clear
shows
that
that
the first
this
inclusion
is also
and let the coordinates
fields
It will
be
P-, is involutive.
the inclusions
in (3.1)
possible
be denoted
X, and X 2. Precisely,
where
d>ix, y) = i/Hx),
ib(x) > 0 for x > 0.
and
by x and
of AI can be joined
curve
of D, it follows
it is clear
that
A^*
A submanifold
fold)
that
A C°° distribution
exists
manifolds
tangent
an integral
property
to every
the integral
as shown
Pp has
A has
dimension
manifolds
by the distribution
was introduced
2
manifold
property
A^,
in the example
two if x > 0.
must
where
submanifold
manifolds
that
X belongs
every
integral
On the other
hand,
(or integral
mani-
S
is exactly
if for every
If A has
ttz £ M
the integral
to A if and only if X is
that
be involutive.
D = iX„
space
property
m £ S.
It follows
discussed
Since
x < 0.
s £ S, the tangent
field
of A.
ifi(x) = 0 for x < 0 and
two everywhere.
5 of A such
then a C°° vector
integral
ex-
M be R ,
of two vector
x > 0 by a piecewise
to be an integral
the integral
manifold
that
dimension
one whenever
A if, for every
A(s-).
D consist
such
(x, y) with
S of M is said
there
Let
Let
X2 = (bd/dy
Ppix, y) has
to a point
has dimension
of the distribution
y.
The following
inclusion.
let
if/ is a C°° function
Clearly,
point
may be proper.
for the second
Xx = d/dx,
that
is involutive.
then
ADÇA^CPD.
ample
has
D^
defined,
a C
distribution
The converse
X2J is the set
above.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
which
is not true,
of vector
fields
178
H.J.SUSSMANN
If A is a C
submanifold
distribution,
integral
(a) 5 is an integral
manifold
of A, and
(b) every
integral
manifold
connected
submanifold
folds
trivially
through
metric
ttz must
A has
ttz £ M there
Remark.
tion
integral
a maximal
integral
passes
of Ap*
y that
pass
Pp.
that
S is an open
to define
through
to T . Therefore,
any two maximal
integral
property
manifold
have
Suppose
the orbits
submani-
that
if through
of clarifying
we are trying
curves.
contained
If X £ D,' then
a distribu-
integral
manifolds
vectors
at ttz of
in the D-orbit
the curve
X(t7z). Now let
the geo-
to define
A(ttz) as the set of all tangent
A(ttz) must contain
every
of A.
the purpose
of D are the maximal
m and are entirely
T ÏT2 be the set of all such
longs
intersects
manifolds
considerations
and
It is reasonable
curves
of A is a connected
coincide.
The following
meaning
that
the maximal
A with the property
of A.
of A which
from the definition
a point
We say that
point
manifold
of A.
It follows
Let
a maximal
S of M such that
of m.
Z —►X t im) be-
X £ D,
Y £ D.
The
curve
t^XtiYtiX_tiY_tim))))
also
belongs
vector
to T.
to this
A(ttz) must also
brackets,
After
curve
a reparametrization,
at Z = 0 is
contain
[X, YKttz).
and we conclude
may be more directions
For instance,
is a curve
XAyit))
that
that
A similar
Ap*
have
to be included
that
belongs
y(0) = ttz'
and
y eV
to T . If v is the tangent
Therefore,
A(ttz).
be D-invariant.
Therefore,
smallest
A must
D-invariant
directly
D-invariant
that
corollary
distribution
the same
distribution
from the results
that
of the next
of Theorem
4.1).
which
result
far away",
ing back.
Directions
of motion
8 given
If t—> yit)
by Z—>
to y at Z = 0, then
then
dXTiv)
suggests
A p*.
of Ap^m).
the tangent
must belong
that we define
It would
to
A to be the
not be hard to
A to be the smallest
Ap,
by taking
A = Pp
i.e.
Specifically,
The reason
why
Ap*
the inclusion
an integral
obtained
in this
(this
will follow
Ap* Ç Pp
may not be the "right"
It may happen
More precisely,
along
those
there
by taking
contains
moving
to higher
However,
is achieved
section.
many directions.
by "going
in A.
the curve
vector
contains
Therefore
Write m' = X_Tim).
then
This
the tangent
can be applied
in A(ttz), besides
if v £ Aim'),
tion to look at can now be understood:
sufficiently
,,
that
[3, p. 97]).
argument
let X e D and let T eR be fixed.
such
known
must be contained
to 8 at t = 0 is dXTiv).
prove
it is well
LX> YKttz) (cf. Helgason
that
Ap*
one may move
curve
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
within
distribu-
not contain
the orbit
of ttz
of an X £ D, and then com-
way need
but they will belong to Ppim).
does
is a
not belong
to Ap*(77z),
ORBITS OF FAMILIES OF VECTOR FIELDS
4. Statement
Theorem
set of C
of the main results.
4.1.
Let
M be a C°° manifold,
vector fields.
is equal
of D, then
structure
to its rank,
such
integral
(c)
Pp
that
as defined
(b) With the topology
maximal
and let
D be an everywhere
S (with the topology
S is a submanifold
defined
of §2) admits
of M.
The
a unique
dimension
of S
in s 3.
and differentiable
submanifold
has
the two main theorems.
Then
(a) // S is an orbit
differentiable
We now state
179
structure
of (a), every
orbit
of D is a
of Pp.
the maximal
integral
manifolds
property,
id) Pp is involutive.
Our formulation
all the relevant
that parts
of Theorem
information
(c) and (d) of Theorem
We now turn to our second
Theorem
M. Let
4.1 was chosen
about
4.2.
Let
D be a set
is everywhere
the orbits
4.1 are immediate
M be a C
Then
manifold,
fields
which
the following
(a) A has the integral
manifolds
(b) A has the maximal
integral
(c)
the statement
will contain
corollaries
Pp.
Notice
of (a) and (b).
theorem.
of C°° vector
defined).
so that
of D and the distribution
and let
A be a C
distribution
spans
A iso that,
in particular,
conditions
on
D
are equivalent:
property.
manifolds
property.
A z's D-invariant.
(d) For every X £ D, t £ R, m £ M such that X im) is defined,
then dX
maps A(ttz) into A(X im)).
(e)
For every
m £ M there
exist
elements
X , • ■• , X
of D such
that
(1) A(ttz) is the linear bull of Xlim), ■■■ , Xk(m), and
(2) for every
f\ (l < z, j < k) which
X £ D there
exists
are defined
c > 0 such
that
in the open-interval
there
i-e,
are
C
functions
e) and satisfy
[X, X'l iXtim))= ¿ fjit)XHXtim))
i- i
for -e< t< e, i = 1, • • • , k.
it) A = PD.
Theorems
5.
Proof
vector
fields
k denote
duced
integers
4.1 and 4.2 will
of Theorem
on M.
the rank
in §2.
n.
Let
Recall
Let
of S.
4.1.
We let
S be an orbit
We shall
D°° denote
that,
be proved
use
two sections.
D be an everywhere
of D.
We give
the notations
the union
if ç £ D",
in the following
then
of the sets
fi¿r
defined
set of C
S the topology
ppb,772 , fi/rb.ZTZ that
D",
taken
is an open
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
over
subset
of §2.
were
intro-
all positive
of R",
Let
and
180
H. J. SUSSMANN
pe
is a C
msp f'om
uous
map into
S.
V(<f, m,T)
Theorem
into
M. Moreover,
if ttz e 5 then
If ¿; e D°°, m £ M, T £ tip m, we let
age of the tangent
that
fit
space
to fi/r
is a subspace
4.1 will
at T under
of pp
the im-
. It follows
ttz' = zf-Grz).
of the following
lemmas.
Lemma 5.1. Let ¿f £ D°°, ttz e S, T £ Up m, mQ = ^t(ttz).
contained
is a contin-
V(¿j, ttz, T) denote
the differential
of M i, where
be a consequence
pp
Then ViÇ, m, T) is
in PpimA.
Lemma
5.2.
Let
77zn £ S.
Then
there
exist
<f £ D°°, m £S,T
efit
szzci
Zzjaz ¿Jt(ttz) = 77zn and V(£, m, T) - PpimQ).
Lemma
5.3.
its underlying
Proof
exist
Let
N be a connected
set of points.
If U intersects
that the lemmas
imply Theorem
m £ S, (fe D00, T eu;
The differential
not exceed
implicit
of pp
function
of PD
S then
U is
Let
m0 £ S.
4.1.
k at T.
of fit
theorem,
submanifold
and let
an open
subset
By Lemma
U be
of S.
5.2 there
such that £Tim) = mQ and V(<f, m, T) = Ppim0).
has rank
k at any point
integral
there
ttz0 in M, and diffeomorphisms
. Let
exist
By Lemma
<f £ D".
neighbourhoods
tf>, if/ from
5.1, the rank of dpp
By awell-known
(i of T in R"
U onto
C"
can-
form of the
and from
and
V onto
V of
C^
such
that the following diagram
P?..m
commutes,
and that
4>iT) = 0, if/im Q) = 0.
C = {(*!,■
E
'0,
vher
Let
E
we are using
the notations
xp) : -1 < x. < 1 for i = 1,
, p\,
p,k{xv ••• >v> = (v •••»**. °» ••• »o).
, 0" denotes
A denote
Here
a string
the submanifold
of p — k zeroes,
of M which
and
p = dim M.
is the inverse
¡AC"). The set of points of A is precisely
pp
((/).
image
under
if/ of
If T £ U, then the
tangent space of A at ttz' = pp (T ) is V(<f, m, T ). By Lemma 5.1, V(<f, m, T )
is contained
A
in Pp(Trz').
, = Ppim').
is contained
Since
Therefore
in S.
both
By Lemma
have
dimension
submanifold
5.3, the set of points
note the inclusion
map from A into
connected
W of A.
subset
spaces
A is an integral
Moreover,
S.
Lemma
k, it follows
of Pp.
of A is open in S.
5.3 can be applied
the open connected
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sets
that
It is clear
that
Let
to every
constitute
A
/ de-
open
a basis
181
ORBITS OF FAMILIES OF VECTOR FIELDS
for the topology
of A.
of the inclusion
/':
feomorphism
we conclude
morphism
Therefore,
A —* V, the diffeomorphism
qb~
and the map
that
/ is continuous.
onto its image
Let
2
denote
each
It is clear
above.
that
a family
S which cover
the sets
/ is
C"".
Since
/ is continuous.
clusions
that
S is connected
that
(cf.
As before,
we can apply
sets
map from
phism.
Let
that
/ is
Lemma
Let
Therefore
Since
Since
/ is regular,
V = T , so that
integral
The preceding
paragraphs
statements
follow
complete,
modulo Lemmas
Proof of Lemma
72= 1, the desired
5.1.
T
that
that the in-
which
submanifold
integral
of points
that
It is clear
submanifold
be the open
subset
that
submanifold
that
of S.
the in-
of 5 whose
The preceding
T are submanifolds
we conclude
subset
of V, and use
It follows
map.
that
of Pn which
of T is an open
of T.
S is
of Pp.
of Pp.
open connected
and
is compatible
of M. The unique-
re-
of M, it fol-
/ is a C°° diffeomor-
submanifold
of S.
The proof that
of Pp is complete.
establish
trivially
statements
from these.
(a) and (b) of Theorem
The proof
4.1.
The
of Theorem
4.1 is now
by induction
on tz. If
5.1, 5.2 and 5.3.
Let
conclusion
Z —>X{m)
of the fact
submanifold
V is an open
submanifold
remaining
V
to prove
if we show
from our construction
for the topology
Let
n U(A)
be denoted
It is sufficient
will follow
/: T' —> T be the identity
/ is continuous.
S is a maximal
tQ to the curve
5.3 to every
S is open.
UiF).
C°°.
It follows
UÍT)
to
of t/(Aj)
manifolds
structure
T be a connected
are a basis
on S in such a way
It is sufficient
S is a submanifold
integral
of
z= 1, 2.
S is an integral
5.3, the set
T into
set is
show
and that
§2).
the fact that these
that
is immediate.
By Lemma
map.
of M, this
There-
of open subsets
structures
a differentiable
5 is a maximal
S.
underlying
S admits
manifold,
intersects
lows that
of S.
of / is a consequence
and is such
of S, and that for
structure
Let these
for
by the construc-
the underlying
onto its image.
on a family
A2 coincide.
S,
/ is a homeo-
U(A) denote
a differentiable
S are homeomorphisms
a structure
We now show
marks
structures
W. —» W2 be the identity
of §2,
a ¿-dimensional
clusion
let
S is a homeomorphism
But the continuity
We have proved
of such
and
that
can be obtained
A £ 2,
W is a submanifold
of W. into
with the topology
ness
A that
shown
^k, tne dif-
in S.
if A. £ S (z = 1, 2) then the differentiable
by W,, W2, and let /:
that
is open
of 2 will be open submanifolds
of A,
E
as a map into
iXA) form an open covering
of A onto
of differentiable
as open submanifolds
/ is the composite
if/, the mapping
we have
image
For each
Moreover,
pp m is continuous
Therefore
this
S. We want to define
that all the members
prove that,
Since
the set of all manifolds
A £ 2 the inclusion
fore, we have
Pp m-
and that
tion that was described
set of A.
/ is an open map.
£ £ D".
reduces
belongs
to
We prove
our result
to the assertion
PpiX(
im))
that
for every
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the tangent
vector
X £ D and every
at
ZQ
182
H. J. SUSSMANN
such
that
XtAm)
definition
is defined.
The
truth of this
assertion
is immediate
given
the
of Pp.
Now let £ £ D", T £ Dn. Write £ = Xrj, T= tQT\ where X £ D, 77£ D*_1,
T' £ fi 17,777,' Z„
£ R.
0
Assume
V(£ Z72,T) is spanned
Since
that
our conclusion
is true for 77.
It is clear
'
by X(£T(m)) and the image under
Vin, m, T1) is contained
in
PpinTAm)),
and
dX,
that
of V(n, m, T').
Pp is D-invariant,
it follows
that
dXt iVi-q, m, T')) Ç Pp(¿; (ttz)).
Also,
it is clear
contained
that
X(£r(m))
in Pp(£T(m)).
Proof
of Lemma
belongs
The proof
5.2.
to Ppi^jim)).
of Lemma
We shall
prove
Therefore
V(¿j, ttz, T) is
5.1 is now complete.
the following
two assertions:
(a) If £ £ DM, m £ S, T £ fi p m, 77e D°°, ttz' £ S, T' £ Q
¿¡Am) = r]T,im')
= mQ, then there
Vi¿j, m, T) and
Vir], m', T')
(b) There
every
is a subset
v £ A there
v £Vi£
It is clear
in
such
that
V(o, ttz", T").
which
spans
Ppim0)
¿j £ D°°, m £ S, T £Ü,p m such
that Lemma
to prove
To prove
and is such that
that
for
¿;Tim) = 7720 and
Then
identity
mapping,
field
It is clear
that
0= <f<5?, T" = Ti-T)T
that
= nTiim')
This
proves
A spans
of o, T , and
to some
Pp(mA.
g= £T (£ £D°°, T efi^J.
Therefore,
= rrz0. Since
Let
Y £ D under
is tne
More-
V(o", ttz", T)
X(t7z0), where
a local
v £ A, and let
Let77=<iX,
of the differential
v £ V(iy, m, T'),
^Tt_T
Vin, m', T ).
ttz that
A to be the set of all vectors
corresponds
the image
Therefore,
also
(a).
and, for some m £ S, g(m) = tt20 and w £ M
Moreover,
5.1.
(here we use the notations
V(a, 772",T") will contain
from the definition
(b), we take
which
from (a), (b) and Lemma
= ¿;ri¿;_Tir)Tiim')))
it is clear
V(<f, ttz, T).
To prove
ttz" = m',
oTJ.m")
it is immediate
contains
5.2 follows
(a) and (b).
(a), take
of §2).
vector
are contained
A of Pp(mQ)
exist
,, are such that
o £ D°°, ttz" £ S, T" £Üo.m„
m, T).
we only have
over,
exist
v = dg(w),
where
is of the form XW,
T' = (T,0).
of p
at
and (b) is proved.
X is the
diffeomorphism
g£Gn.
g £ Gp
X £ D. Let
Then r¡T,im) = gTim) = mQ.
T' certainly
The proof
contains
of Lemma
d£Tixim)).
5.2 is now com-
plete.
Proof of Lemma 5.3.
Let
5) be the set of all vector
fields
that are of the form
dgiX), for some X £ D, g £ Gp. Let Y e ®. Let Y = dg(x), g £ Gp, X £ D. If
y is an integral
X.
of Y, then
From this it is clear
Let
•••
curve
N be a connected
, Xp be elements
y is the image
that any two points
integral
of 3) such
that
g of an integral
in y are in the same
submanifold
[xAm),
under
of Pp,
•••
and let
, X^ttz)!
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curve
of
D-orbit.
m £ N.
is a basis
Let
X1,
for PpU).
183
ORBITS OF FAMILIES OF VECTOR FIELDS
The mapping
it., ...,
is a diffeomorphism
N.
to the same
borhood
which
conclude
that
imply
m.
as
t —►(r,,
contained
that
that
that,
of some
and Lemma
every
in one orbit of D.
in S.
We must
of R".
that,
of tQ.
of this
point
Since
implication
that
mapping
be-
a neigh-
N is connected,
in an orbit
of D.
show that,
we
If U O S
if
if m £ S, ç £ D",
it follows
in
p¿
that
yit)
pp
(d) => (c) =» (f)
and
(U)
If y
£ U for all
the image
Therefore
(f) => (b) follows
Assume
that
of the
If í > 0 is small
From
We now show
of the definition
smooth
Therefore
their
coefficients,
enough,
then
the curve
that
(a) =» (e).
The
following
that
(e) =» (d).
bracket,
but we state
Let
under
is open,
We now show
lemma
the vec-
to S are linear
neighborhood
is a trivial
because
m £ M
, X^ be elements
Z —» A' (m) is contained
it separately
are
that
Let
If X £ D then
restrictions
in some
(b) => (a)
property.
X , ...
for A(ttz).
it is immediate
of Lie
4.1.
manifolds
m.
form a basis
to S.
X1 with
from Theorem
the integral
of A through
• • • , Xkim)
\t\ < e.
this
A has
manifold
xAjn),
(in S).
U oí m
in
U for
consequence
we shall
need
later.
Lemma 6.1. Let X and Y be C°° vector fields
on M. Let m £ M, and let
c > 0 be such that X Am) is defined for \t\ < c. Let
W(t) = dX_t(Y(Xt(m)))
Wit) £ M . Then
Wit) satisfies
for \t\ < (,
the differential
equation
VU) = dX^([X, Y](Xt(m)))
with initial
(T)
of any
of an X £ D.
ZQ, then
easily
U.
under
curve
y(tQ) £ U for some
Prom this
be such that pp
1 < i < n, the image
The implications
[X, Xz] are tangent
combinations
so that
of ttz in
of N has
Let T £ fi ¿ m (C P")
of T is contained
4.2.
and let S be an integral
it again
in the image
that
t, T. ., •• • , r ) is an integral
of Theorem
The
tor fields
point
shown
onto a neighborhood
5.3 is proved.
(a) =» (e) => id).
of D such
every
i such that
it follows
neighborhood
6. Proof
trivial.
(ttz)... ))
i
U C S.
for each
• • • , r.,,
image,
(...x?
of N is contained
U is open
Z in some neighborhood
Pp
that
ill) is an open subset
is one such
i2
of 0 in Pp
We have
U of points
it follows
£ U. It is clear
curve
remarks
is entirely
We now prove
then pp
tx
orbit
the set
is nonempty,
P
from a neighborhood
The preceding
longs
t) -xMx,2
condition
W(0) = Y(m),
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H. J. SUSSMANN
184
Now assume
that
condition
(e) holds.
To prove
for \t\ < t, e > 0. Let X1, ••• , Xk, e, (fi).
Let Wl(t) £M be defined by
(d), it is sufficient
to do it
.= J . , , fc be given by condition
WAt) = dX_t(XiiXl(m)))
(e).
for \t\ < e.
By Lemma 6.1, we have
dWÀÙ/dt= dX_{[X, X*KX,(in))).
The right
side
of this
equation
is equal
to
X f)i(t)dX_tiXAXim))).
7=1
Therefore,
the
W1 satisfy
the system
of differential
equations
Ädt = ff* fi.w.
']
7=1
Since WHO), ■■■, Wki0) are a basis for Mm), it follows that wHt), ■■■,
Wk(t)
form a basis
for A(ttz) for -e < Z < e. Since
dXt(Wi(t))= XÁXt(m))£A(Xt(m)),
we conclude
that
Remark.
dX
The preceding
7. A theorem
a manifold
actly
of Chow.
M is said
the connected
and sufficient
Proof.
M. Then
has dimension
of Pp are precisely
the reachability
is less
of S is empty.
than
Corollary 7.2.
defined
D satisfies
n for every
the connected
condition.
72 for some
connected
Then
set
in Lobry
[5].
D of C°° vector
condition
of M. The following
D be an everywhere
C ■ manifold
ze-dimensional
defined
the reachability
If dim Ppim) = 72 for every
D satisfies
of Ppim)
An everywhere
contained
is complete.
if the D-orbits
theorem
gives
fields
on
are ex-
a necessary
for reachability.
Let
and only if Ppim)
A(X (ttz)), and our proof
proof is essentially
components
7.1.
n-dimensional
A(ttz) into
to satisfy
condition
Theorem
folds
maps
submanifold
the reachability
set
of C
vector
the reachability
on the
condition
if
m £ M.
m £ M, then the maximal
components
Conversely,
m £ M. Then
of M, and
condition
fields
integral
submani-
of zM. By (b) of Theorem
suppose
the orbit
k < n.
4.1,
that the dimension
S of D through
Therefore
k
ttz is a
the zM-interior
is not satisfied.
Let M, n, D be as in Theorem 7.1.
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Let Ap*
be the dis tribu-
ORBITS OF FAMILIES OF VECTOR FIELDS
Zz'o72spanned
fies
by D . If Apjm)
the reachability
Proof.
every
that
X £ D
dimension
n for every
m £ M, then
D satis-
condition.
Every
Ppim)
has
185
X £ D belongs
belongs
to Pp.
has dimension
to Pp.
Since
Therefore
Pp is involutive,
Ap* Ç Pp.
72 for every
it follows
Our assumption
m £ M. The conclusion
then
that
implies
now follows
from
Theorem 7.1.
The preceding
condition
corollary
for reachability
example
is obtained
Ap*(?7z) has
ample
shows
have
maximal
D consist
given
in §3.
of three
example,
dimension.
it is possible
dimension
0(x,
fields
y, z) = p(x)
finitely differentiable
One sees
dimension
that,
however,
there
slight
and
even
X2, X,,
which
the sufficient
necessary.
X2I of vector
ttz £ M. We take
X,,
that
fields
A simple
in the plane
are points
ttz such
modification
when
of the ex-
Ap*(m)
does
not
M = P , and we let the set
are defined
as follows:
X2 = (bd/dy,
Xi = xbd/dz.
ifiix, y, z) = oix).
The functions
p and
o are in-
and satisfy
p(x) = 0
for x < 0,
oix) = 0
for x > -1,
pix) > 0
for x > 0,
oix) > ft
for x < -1.
condition
is satisfied.
easily
that
of Ap*(x,
Theorem
duction
is by no means
iXj,
to have reachability
Xx = d/dx,
Here
We emphasize
The following
at any point
vector
[2].
result
the pair
In this
maximal
that
in this
by considering
that was introduced
that
is due to Chow
the reachability
y, z) never
exceeds
7.1 and the preceding
in general,
Ap*
However,
the
two.
example
justify
is not the "right"
the claim
distribution
made
that
in the introis needed
for
the study of D-orbits.
8.
which
pose
Integrability
it has
been
is to show
type
that
how the known
An everywhere
finite
of distributions.
proved
defined
if for every
set
We now discuss
a distribution
results
has
follow
exist
conditions
submanifolds.
under
Our pur-
from ours.
D of C°° vector
m £ M there
various
integral
vector
fields
is said
to be locally
fields
X , • >• , X
which
of
belong
to D and satisfy
(LFT1) xHttz), • • • , X^Httz)span ApU),
(LFT2)
tions
for every
f1., defined
X £ D there
exist
and
a neighborhood
on (7, which satisfy
[X, X'Kttz')= ¿
7= 1
fKm')X>(m)
'
for all ttz' £ U.
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II of m and
C°° func-
186
H. J. SUSSMANN
Theorem
manifolds
8.1.
// D z's locally
Proof.
Theorem
If D is locally
that
D
manifolds
The
(with
result
A = Ap).
of finite
property.
then
Ap
has
the maximal
integral
Theorem
Let
type,
integral
theorem
8.2.
it is clear
Therefore,
that
4.1 and 4.2,
manifolds
of Frobenius
of m.
condition
our conclusion
[4] and Lobry
it follows
(e) of
follows
from
Then
Ap*
Ap*
[5].
If D C VÍM) is
has
the maximal
inte-
= Pp and the D-orbits
of Ap*.
can now be proved
A be a C°° distibution
of A(ttz) is independent
that
(a) of the same theorem.
By Theorems
the maximal
classical
then
is due to Hermann
is locally
are precisely
erty
type
type
of (e) with condition
The preceding
gral
of finite
4.2 is satisfied
the equivalence
such
of finite
property.
on M such
A has the maximal
easily.
that
the dimension
integral
manifolds
k
prop-
if and only if it is involutive.
Proof.
We know that if A has the maximal
is involutive.
Conversely,
X , ••• , X
be
let us assume
k C°° vector
• ■ • , X (ttz) form a basis
X (ttz') are linearly
mension
vector
field
that
belongs
that
fields
which
for all
in some
combination
to A, the preceding
[X, X'],
which
of finite
that
belong
type.
that
X (ttz),
of ttz. Since
the di-
of m, every
of X , ■ • • , X
conclusion
applies
with
smooth
in particular
A is involutive.
The conclusion
A
X (ttz'), • • ■ ,
neighborhood
to A because
then
ttz e M. Let
to A and are such
we can assume
that,
property
Let
m' in a neighborhood
to A is a linear
A is locally
manifolds
A is involutive.
belong
Clearly,
k, it follows
If X belongs
to the vector
shown
independent
of A(?T2') is always
coefficients.
fields
for A(ttz).
integral
that
now follows
We have
from Theorem
8.1.
In particular,
by an involutive
Corollary
sume
has
that
Let
set
8.3.
D
Let
D
applies
By Theorem
be an everywhere
and that
integral
ttz £ M and let
result
manifolds
8.2,
A„*
spanned
it will
be sufficient
borhood
U of ttz and are linearly
independent
near
Then
X and
defined
set
of C
vector
dimension
k.
fields.
Then
As-
Ap*
property.
be elements
cients.
A is the distribution
has constant
X , ••• , X
m.
when
D :
is involutive
the maximal
Proof.
the preceding
X' are linear
to show
of D
which
at m.
combinations
Let
of the
that
Ap*
are defined
X and
X1 with
X
is involutive.
on a neighbelong
smooth
to Ap*
coeffi-
The formula
[fY, gZ] = (/ . Yg)Z- (g . Zf)Y + fg[Y, Z]
together
with the fact that
the brackets
[X\
X7] are elements
[X, X'] belongs to A.
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of D , implies
that
ORBITS OF FAMILIES OF VECTOR FIELDS
We now show
considers
that
our results
a Lie algebra
also
contain
L of (everywhere
those
187
of Matsuda
defined)
C°° vector
[6].
Matsuda
fields
on the mani-
fold M. Let us define inductively
[X(0), Y] = Y,
for any two
C°° vector
(C) For every
fields
X and
[X^+1), Y] - [X, [X(H Y]]
X and
Y on M.
Matsuda's
Y in L and for every
oo
i-Dkl-[X^\
k =0
fines
this
for (z, ttz) in some
a continuously
function
differentiable
For each
Theorem
Proof.
of (0, ttzq).
function
by termwise
8.4.
// the Lie algebra
m —> L(t7z) has
In view
of Theorem
ttZq £ M, and X (ttZq) is defined,
gAX,
Y)(ttz) de-
of Z and
m, and the derivatives
of the series.
of
We prove:
L satisfies
the maximal
4.2,
Moreover,
differentiation
m £ M, let L(7tz) = {XÍttz) : X £ L\.
the distribution
yKttz)
k-
neighborhood
can be obtained
is
k
gt(X, Y)im)= £
converges
condition
mQ £ M, the series
Matsuda's
integral
it is sufficient
condition
manifolds
to prove
(C), Z¿e?2
property.
that,
if X £ L,
then
dXt(L(mQ))Ç L(X(U0)).
Clearly,
it is sufficient
to show that,
L(Xt(mQ)) for sufficiently
small
if v £ L(m0),
then
dX((v)
Z. Let v = Y(mQ), where
belongs
Y £ L.
to
Let e > 0 be
such that
oo
k
Vit, r)= ¿ (-1)* i- [xAk\ Y](XrUn))
fe=o
*•
converges
dX_
for
\t\ < e, \t\ < c, and can be differentiated
(V(z, r)) so that
with respect
W(t, r) e Mm .
to r of dX_Ti[X{k),
It follows
term by term.
from Lemma
6.1 that
Let
W(t,r)
=
the derivative
Y](Xr (ttz))) is dX_r [X{k + 1 \ Y](Xr (ttz)). From
this it follows immediately that dW/dt + dW/dr = 0.
In particular,
the function
Z —> Wit, t) is a constant,
so that
Wit, t) = v for
all Z.
Therefore,
clear
that
dX iv) - Vit, t).
Vit, r) belongs
and our theorem
9.
tions
stronger
L is a Lie algebra
to L(Xr(772)) for every
t, r.
of vector
Therefore
fields,
it is
dXtiv) £ LiX(im)),
is proved.
The analytic
of .§2
Theorems
Since
and §3
case.
If M is a real
can be reformulated
with
4.1 and 4.2, and the consequences
conclusion
that the orbits
analytic
manifold,
"C°°"
of §7
replaced
and
of Pp are analytic
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then
all the defini-
by "analytic".
§8 remain
submanifolds
valid,
with the
of M. How-
188
H. J. SUSSMANN
ever,
there
is an additional
analytic
vector
the ring
A^
fields
noetherian
•••
of analytic
of vector
such
that
makes
D
everything
is locally
functions
fields
as an A^-module.
, Xp of D
written
that
on M, then
of germs
the set of germs
°
fact
of finite
at a point
every
X £ D
(and,
of ttz as
type.
This
A".m
is so because
It follows
are finitely
in particular,
£?_.
// D z's a set of
ttz £ M is noetherian,
at ttz is the rproduct
If ttz e M, then there
in some neighborhood
simpler:
A"m is
many elements
every
f X1, where
that
and
[X, X!])
the functions
f
X ,
can be
are
analytic.
In view of Theorem
folds
property.
This
8.1, the distribution
result
was proved
Ap*
has
by Nagano
[7].
the maximal
The method
integral
mani-
of proof
based
on Theorem 8.1 is due to Lobry L5J«
REFERENCES
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Theory
Princeton,
2. W. L. Chow,
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N. J., 1946.
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Ordnung, Math. Ann. 117(1939), 98-105.
3. S. Helgason,
Differential
I, Princeton
Math.
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MR 7, 412.
partiellen
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erster
MR 1, 313.
geometry
and symmetric
spacer,
Pure
and Appl.
Math.,
vol. 12, Academic Press, New York, 1962, MR 26 #2986.
4. R. Hermann,
Nonlinear
On the accessibility
Differential
Equations
1963, pp. 325-332.
5. C. Lobry,
573-605.
problem
and Nonlinear
in control
Mechanics,
theory,
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Academic
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Sympos.
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York,
MR 26 #6891.
Controlabilite
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SIAM J. Control
8 (1970),
MR 42 #6860.
6. M. Matsuda,
An integration
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larities, Osaka J. Math. 5 (1968), 279-283.
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Linear
differential
integrable
systems
with singularities
transitive Lie algebras, J. Math. Soc. Japan 18 (1966), 398-404.
8. H. Sussmann
and V. Jurdjevic,
systems
with
singu-
MR 39 #4876.
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MR 33 #8005.
systems,
J. Differential
Equations 12 (1972), 95-116.
DEPARTMENT OF MATHEMATICS,UNIVERSITY OF CHICAGO, CHICAGO, ILLINOIS 60637
Current
address:
Department
of Mathematics,
Rutgers
Jersey 08903
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University,
New
Brunswick,
Ne