Advances in Pure Mathematics, 2011, 1, 245-249
doi:10.4236/apm.2011.15044 Published Online September 2011 (http://www.SciRP.org/journal/apm)
Copyright © 2011 SciRes. APM
Interior and Exterior Differential Systems for Lie
Algebroids
Constantin M. Arcuş
Secondary School CORNELIUS RADU”, Rădineşti Village, Gorj County, România
E-mail: c_arcus@yahoo.com, c_arcus@radinesti.ro
Received March 26, 2011; revised August 9, 2011; accepted August 20, 2011
Abstract
A theorem of Maurer-Cartan type for Lie algebroids is presented. Suppose that any vector subbundle of a Lie
algebroid is called interior differential system (IDS) for that Lie algebroid. A theorem of Frobenius type is
obtained. Extending the classical notion of exterior diffential system (EDS) to Lie algebroids, a theorem of
Cartan type is obtained.
2000 Mathematics Subject Classification: 00A69, 58A15, 58B34.
Keywords: Vector Bundle, Lie Algebroid, Interior Differential System, Exterior Differential Calculus,
Exterior Differential System
1. Introduction
Using the exterior differential calculus for Lie algebroids
(See [1,2]) the structure equations of Maurer-Cartan type
are established. Using the Cartan’s moving frame
method, there exists the following
Theorem (E. Cartan) If n
NMan is a Riemannian
manifold and i
i
XX
x

, 1, n
is n ortonormal
moving frame, then there exists a collection of 1-forms
, ,1,n

uniquely defined by the requirements

and
,1,
F
d

n
where
,1,n

is the coframe. (see [3], p. 151)
We know that an -dimensional distribution on a
manifold is a mapping defined on , which
assignees to each point
r
NDN
x
of an -dimensional
linear subspace
N r
x
D of x. A vector field TN
X
be-
longs to if we have
D
x
x
X
D for each
x
N
.
When this happens we write

X
D .
The distribution on a manifold is said to be
differentiable if for any
DN
x
N there exists differen-
tiable linearly independent vector fields
r

1,,
r
X
XD in a neighborhood of
x
. The distri-
bution is said to be involutive if for all vector fields
D
,
X
YD we have

,
X
YD .
In the classical theory we have the following
Theorem (Frobenius) The distribution is involut-
ive if and only if for each
D
x
N
r
there exists a neigh-
borhood and
Un
linearly independent 1-forms
1,,
rn
on U which vanish on and satisfy the
condition
D
1,rn

,1,
F
drn

 
for suitable 1-forms
, ,1,n

.(see [4], p. 58)
Extending the notion of distribution we obtain the
definition of an IDS of a Lie algebroid. A characteriza-
tion of the ivolutivity of an IDS in a result of Frobenius
type is presented in Theorem 4.7.
This paper studies the intersection between the ge-
ometry of Lie algebroids and some aspects of EDS. In
the classical sense, an EDS is a pair
,
M
I consisting
of a smooth manifold
M
and a homogeneous, differen-
tially closed ideal
I
in the algebra of smooth differen-
tial forms on
M
. (see [5,6]) Using the notion of EDS of
an arbitrary Lie algebroid


,,, N
FN Id
,, ,
F we
obtained a new result of Cartan type in the Theorem 5.1.
In the particular case of standard Lie algebroid
,Id,,
,, ,Id
MTMM
TM there are obtained
similar results those for distributions.
TM M
We know that a submanifold of is said to be
S N
C. M. ARCUŞ
246
integral manifold for the distribution if for every
point
D
x
N,
x
D coincides with
x
TS. The distribution
is said to be integrable if for each point
D
x
N
there exists an integral manifold of containing
D
x
.
As a distribution is involutive if and only if it is
integrable, then the study of the integral manifolds of an
IDS or EDS is a new direction by research.
D
2. Preliminaries
In general, if is a category, then we denote CC the
class of objects and for any ,
A
BC, we denote
the set of morphisms of
,
BCA
A
source and
target. Let
B
lgeA ,Li ,
M
od and be the category of
Lie algebras, modules and vector bundles respectively.
v
B
We know that if

,π,v
EMB,
and
, then

πEM

Ma


,,πanMEu
FM n

,π
,u
M
:
M
Id
,,EM
M
,R

,

is a
F
M-module.
We know that a Lie algebroid is a vector bundle

,, v
F
NB
so that there exists




,,,,
v
N
,,
N
I
dB

FN N

TN
and also an operation




,
,, ,,,,
,,
F
F
F
NFN FN
uv uv
 
 
with the following properties:
1
LA . the equality holds good

N
d


,,,
FF
uf vuvIu f v
 f
,
N
for all and

, ,uv F
f
FN

,
2
LA . the 4-tuple

,, ,,,,
F
FN
 is a Lie

F
N
LA
-algebra,
3. the
M
od -morphism

,
N
I
d
is a
-morphism of
lgALie


,,,, ,,
F
FN
 source and


,,
NN
,

,,,
TN
TN target.
Let
, ,Id

,,FN,,
F
N be a Lie algebroid.
Locally, for any ,1,p

, we set ,F
tt L
t
 



.
We easily obtain that ,L
L

 for any ,, 1,p.

The real local functions ,,,
1,Lp


are called
the structure functions.
We assume that
,,
F
vN is a vector bundle with
type fibre the real vector space and
structure
group a Lie subgroup of
p
R
,,
,p,RGL . We denote
,
i
x
z
the canonical local coordinates on
,,
F
vN ,
where and
1,in1, .p

,
i
Consider
,
i
x
zx
z

a change of coordi-
nates on
,,
F
vN . Then the coordinates z
change to
'
z
according to the rule:
z
If
,,ztF N
 is arbitrary, then


 
,i
Ni
f
I
dztfx zx
x









(2.2)
for any
f
FN and
x
N
.
The coefficients i
change to i
according to the
rule:
i
ii
i
x
x



 , (2.3)
where 1.



The following equalities hold good:
 
,
ii
ii
f
f
fFN
xx







 (2.4)
and
.
kk
ki j
i
Lj
x
x
 
 


(2.5)
3. Interior Differential Systems
Let
,,,, ,,N
FN Id

F
Definition 3.1 Any vector subbundle

of
the vector bundle
be a Lie algebroid.
,π,EM
,,
F
vN will be called interior dif-
ferential system (IDS) of the Lie algebroid
,N
Id,,,, ,FN

F
Remark 3.1 If
.
,π,EN is an IDS of the Lie alge-
broid
,,FNd
,, ,,
FI
N then we obtain a vector
subbundle
00
,N,πE of the dual vector bundle
**
,,
F
N

so that
 
00
**
,π,
,, :0,,π,.
EN
FNSS EN

 



The vector subbundle
00
,π,EN
will be called the
annihilator vector subbundle of the IDS .

,π,EM
Proposition 3.1 If
,π,EN
is an IDS of the Lie al-
gebroid
,,,, ,,N
F
F
N
I
d so that
,π,EN
z


 (2.1)
1,, ,
r
SS then it exists
**
1,, ,,
rp
F
N





linearly independent so that
00 1
,π,,,
rp
EN
 .
Definition 3.2 The IDS of the Lie alge-
broid
,π,EN
N
Id
,,,, ,,
F
FN
will be called involut-
ive if
,,π,STE N ,
F for any
,,π,STE N .
Proposition 3.2 If
N,π,E is an IDS of the Lie al-
gebroid
,, ,FN

, ,,
FId
N and is
a base of the

1,, ,
r
SS
F
N-submodule
then

,π,,EN

,,
,π,EN is involutive if and only if
,π,,E N,SS 
ab
F for any ,1,ab r.
Copyright © 2011 SciRes. APM
C. M. ARCUŞ247
4. Exterior Differential Calculus
Let

,,,, ,,N
F
FN Id


,,
q
be a Lie algebroid. We
denote
F
N
. If
,, q
the set of differential forms of de-
gree q
0q
,,
F
NF
,,zF

N
, the
-
tio
Definition 4.1 For any
n we obtain
the exterior differential algebra


,, ,,,FN
.

N the applica
n
 
,,,, ,
z
L
F
NF


N
defined by
for any



,
zN
LfId z f



,

f
FN and
1q
for any





1
1
1
,, ,,,
,,,,, ,
zq N
q
iq
F
i
Lz zIdzz z
zzz z






,,
q
F
N


e covariant Lie
and , is
e 4.1 If ,

1,, ,,
q
zz FN

tive with respect to called thderivathe
section z.
Theor m

,,zFN

,,
q
F
N


and

,,
r
F
N

 , then
.
zz
LL z
L


(4.1)
Definition 4.2 If , then the a
q

,,zFN
 pplication
 
 
1
,, ,,
,, ,,
z
i
q
z
FN FN
F
NiF

 

 
N
defined by
0
z
if, for any

f
FN and

22
,, ,,,
z
qq
iz zzzz


,
for any , is called the interior

2,, ,,
q
zz FN

sociated to the sectionproduct as .
z
Theorem 4.2 If

,zF
 ,N, then for any

,,
q
F
N
 and
,,
r
F
N

we obtain the
equality

1.
q
z
zz
ii i
 
 
(4.2)
Theorem 4.3 For any we

,,,zvF N
 obtain

,.
F
vz zzvz
L i Li i (4. 3)
Theorem 4.4 The application

F
d
q

1
,, ,,
q
F
F
NF
d
N

defined by
 
,
F
z
dfzL f
for any and

,,zFN







01
01
1
01
,,,
ˆ
1, ,,,,,
ˆˆ
1, ,,,,,,,,
q
qi
Ni i q
i
ij
ijijq
F
ij
zzz
Idzz zzz
zzz zzzz





 

 
F
d
for any
01
,, ,,,
q
zzzFN
,
property:
is unique having the
. (4.4)
This application is called the exterior dif
op
following
F
Ldi

,,,
F
zzz
id zFN

ferentiation
erator of the exterior differential algebra of the Lie
algebroid
,,,, ,,N
F
FN Id

.
Theorem 4.5 The exterior differentiation operator
F
d given by the previous theorem has the following
perties:
1) For any
pro
,,
q
F
N

 and
,,
r
F
N


we obtain
F
d
 
1.
q
F
dF
d
 
 
(4.5)
2) For any
,,zFN
 we obtain
.
FF
z
z
d L Ld (4.6)
3) 0.
FF
dd
orem 4.6 (oThe f Maurer-Cartan type) If
,,,, ,,N
F
FN Id

is a Lie algebroid
ation operator of the exte
differential
and F
d
rioris the exterior differenti
F
N-algebra


,, ,,,FN
, then
we obtain there equationsn type structu of Maurer-Carta
1,1,
F
dtLt tp

 (MC)
2

1
and
,1,
Fi i
dxt in
 . (MC
2)
where
,1,tp
is the coframe of the vector bundle
,,
F
N
quatio called the structure equations
.
These ens will be
of Maurer-Cartan type associated to the Lie algebroid
,,,, ,,N
F
FN Id

.
Proof. Let 1, p
be ar. Since bitrary
,, ,1,dt L



Fttp

it results that
.
F
dtLt t
 



(1)
Since LL


and tt tt


, for any
,1,p
hat
, it results t
1
Lt t
 
 ,1
,.
2Lt tp
 
 

 
(2)
Copyright © 2011 SciRes. APM
C. M. ARCUŞ
248
Using the equalities (1) and (2) it results t
equation (MC).
Let
he structure
1
1,in be arbitrary. Since

,1,
Fi i
dx tp



it results the s2). q.e.d.
Remark 4.1 In the particular case of te standard Lie
algebroid
tructure equation (MC
h


,,,, ,,TNNId Id
NTNN
TN we obtain
,1,
TN ii
dx dxin, (MC
2)'
where
dx vector bund
,1,
ii n is the coframe of thele
and , for all

,,
N
TN N
.
As TN
dd0.
TN 0
i
L
jk ,, 1,ijk n
we obtain

11,.
i k
jk
Lx in 0,
2
Fi j
d dxdxd  (MC
1)'
These equations are the structure equations of
Maurer-Cartan type associated to the stand
gebroid
ard Lie al-



,,,, ,,TNNId Id
.
NTNN
TN
Theorem 4.7 (of Frobenius type) Let

,π,EN be an
IDS of the Lie algebroid


,,,, ,,
F
FN Id

. If
1,
r
N
p
is a base of,
F
N submodule

00
,π,,,EN, then the IDS

,πolut-
ive if and only if it exists
 ,EN is

1,,
inv
F
N
 ,
,1,rp

 so that
1,
,.
F
rp
d p



,1r

Proof. Let is a base of the

1,,
r
SS

F
N
Let
-submodule
that
,πEN

,F



,,,.

,N so
1rp
,,, ,,
rr
SS
 of the

,,SS

11p
S is a baseS
F
N-module


,, ,,FN
.
**
1r
Let

,, ,,
F
N



so that


11rr p
is a base of the

,, ,,, 

F
N-mod-
ule
For any
**

,, ,,FN



.


,1,ab r and ,1,rp

 whe e have t
lities:
,
equa
 
 
0, .
aa
bbb
a
SS
SS
,0

 

We remark that the set of the 2-forms
,1
, ,1,ab rrp
,,,;
aba b

 
is a base of the

 
F
N
have
,
,
Fbcb
bc b
bc b
dAB
C






(1)
where, ,
bc b
A
B
and C
, ,, 1,abc r,
,, 1,rp


bc cb
are real local functions so that
A
A
and CC


.
ula Using the form
  





,,
,,
F
bcN bc

Nc b bc
F
dSSIdSS
Id S

 

SS S
that
(2)
we obtain
,,
bc F
SS
bc
A
 (3)
for any ,1,bc r and 1,rp
 .
,π,EN is an involutive IDS of the We admit that
Lie algebroid

,, N
Id

,, ,,
F
FN . As
,N, for a,,πEny
bc
F
SS ,bc1,r, it results that
,0
b
Sc
F
S
-module
Therefore, we


2,, ,,FN
.
for any ,1,bc r and 1,rp
 .
Therefore, for any ,bc1,r and 1,rp
 , we
obtain 0
bc
A
and
,
1
2
1
b
b
2
Fb
b
dBC
b
BC
 



 

 
As
 
 



1
1
b,,
2
b
BC FN
 
 
 , for any
,1rp


Conversely, we adm
,it results the first implication.
it that it exists
1,,
F
N
 ,
,1rp

, so that
1,
,
F
rp
d



(4)
for any1, .p r
(4) we obtain thaUsing the affirmations (1), (2) andt
0
bc
A
, for any ,1bc1,rp
 .
,
r and
Using the affirmation (3), we obtain
,0
bc
F
SS
for any ,1,bc r and 1,rp
 .
Therehafore, we

,,π,
bc
F
SSE N
ve , for any
,1,bc r. Using thPr the sec-
on
5. Exterior Differential Systems
Le
e oposition 3.2, we obtain
d implication. q.e.d.
t
d
efinit
,,,, ,,N
F
FN I

oid.
ion 5.1 Any ideal

,,I of the exterior dif-
ferential algebra of the Lie algebroid
be a Lie algebr
D
Copyright © 2011 SciRes. APM
C. M. ARCUŞ
Copyright © 2011 SciRes. APM
249



,,,, ,,N
F
FN Id

closed under
differentiation 1,,
rp
be a base of
operator
F
d, namely
of the Lie algebroid
F
dI I
, is called differential
ideal

,,,, ,,N
F
N Id


. F
be aDefinition 5.2 Let
,,I differential ideal of
the Lie algebroid

,,,, ,,FN I

for all kN
and
,,
N
d. If it exists
F
an IDS

,π,EN so that

k
I
F N

 we

1,, 0
k
uu
, for any

1,,,π,
k
uu EN, the
have
n we will say that
,,I
is
an sys exterior differential (EDS) of the Lie algebroid

tem

,,,, ,,
F
FN

Theo of Cartan type) S

,πE

N
Id .
rem 5.1 (
the Lie algebroid
The ID,N of


,, N
Id is
by
,, ,,FN

involut-
f the ideal generatedthe

F
ive, if and only i
F
N-submodule


00
,π,,,EN is an EDS of the
Lie algebroid

,N
Id .
,

,, ,FN,
F
Proof. Let
,π,Ee Lie
algebroid

N
be an involutive IDS of th


,,,, ,,N
FN Id

. Let

1,,
rp

se of the
F
be a ba
F
N-su
0
π,
;,
N
bmodule

,


,,
q
00
,π,,,EN.
We know that


0
,IE
1rp
qN
F
N
.

,,
q

Let qNand 1
 

,,
rp
F
N
 be
7 we obtain

1
1q
q
bitrary.
Using the Theorems 4.5

FF
F
dd
d

and 4.

1
ar-
F
d
.



 

it
 
 
sults that

As
 
12
1,,
q
Fq
dF

N
re-
00
,π,EN.


00
,,π,.I EN
F

Therefore, ,
F
dI E
sely, let
dI 


00
πN
Conver
,π,EN
broid

be an IDS of the Lie alge-


,N
Id,,,, ,
F
FN

so that the
F
N
-submodule 00
,πIE
alg


,,N

,
is an EDS of the Lie
ebroid
,

,,, N
FN Id

.
Let the

F
N-submod-
ule
,
00
,π,,NE
 .
As

00
,,π,NI Eit results
that
it
00
,π
F
dI EN
exists
1,,
F
N
 , ,1rp

, so that

.
Using the T r

00
1,
,π,
F
rp
dIEN


 
heorem 4.7 thereesults that
,π, is
EN
an involutive IDS. q.e.d.
6.
ae International Foundation
t Tokai University, during
ould also like to thank Pro-
rbanski, “Lie Algebroids and Pois-
Structures,” Reports on Mathematical Phys-
. 2, 1997, pp. 196-208.
doi:
Acknowledgments
I would like to thank Matsum
for the research grant a
April-September 2008. I w
fessors Hideo SHIMADA and Sorin Vasile SABAU
from Tokai University-Japan, for useful discussions and
their suggestions. In memory of Prof. Dr. Gheorghe
RADU. Dedicated to Acad. Prof. Dr. Doc. Radu MIRON
at his 84th anniversary.
6. References
[1] J. Grabowski and P. U
son-Nijenhuis
ics, Vol. 40, No
,,
F
10.1016/S0034-4877(97)85916-2
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L. I. Nicolescu, “Lectures on the G[3] eometry of Mani-
folds,” World Scientific, Singapore, 1996.
doi:10.1142/9789814261012
[4] M. de Leon, “Methods of Differential Geometry in
Analitical Mechanics,” North-Holland, Amsterdam, 1989.
[5] R. L. Bryant, S. S. Chern, R. B. Gardner, H.
schmidt and P. A. Griffiths, “
L. Gold-
Exterior Differential Sys-
tems,” Springer-Verlag, New York, 1991.
[6] T. A. Ivey and J. M. Landsberg, “Cartan for Beginners:
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Providence, 2003.