Pub
.
Ma
.
UAB
ol
.
30
nó
1
Maig1986
ON
HOMOGENITY
AND
TRANSITIVITY
OF
FIELDS
OF
GEOMETRIC
OBJECTS
Je zy
Konde ak
ABSTRACT
.
I
a
is
a
ield
o
geome ic
objec son
amani old
M
hen
we
can
associa e
wi h
i
a
p incipalsubbundle
o
H
(M)
.
We show
ha
(in ini esimal)
homogeni y
and
(iñ ini esi
mal)
ansi i i y
o
his
subbundle
a e
equi alen
o
some
in _e
g alcondi ions
o
he
Lie
eaua ions
gene a ed
by
a
.
I
.
INTRODUCTION
l
.
Le
M
be
a
di e en iable
mani old
.
In
he
p esen
pape ,
mani olds,
ec o ields
and
so
on
always
mean
di e en-
iablemani olds,
di e en iable
ec o
ields
and
so on
.
Di e
en iabili y
always
means
hedi e en iabili y
o
class
I
U,V
a e
open
subse so
M
hen
a di eome phism
U
-
V
is
called
a
lccal
di eómo phism
o
M
.
These
(M) o
all
local
di eomo phisms
o
M
is
a
pseudog oup
.
By
TPA
we shalldeno e
a
se
o ec o
ields
de ined
on opensub-
se s
o
M
.
We
deno e
by
H
(M)
he se
o
all
-je s
a
0
o
di-
di eomo phisms
o
openneighbou hoods
o

0

in
1R
n

on o
open
subse s
o

M
.

Le

i
:
H
(M)
-+
M

be
he
a ge
p o-
jec ion
.
Then
H
(M)

is
ap incipal
ib ebundlewi h
he
s uc u e
g oup
L
o
all
-je s
wi h
he
sou ce
and
wi h
he
a ge
a
0
o
local
di eomo phisms
o
IR
n
.
2
.
Le
F
deno e
a
na u al
bundle
om
he
ca ego y
o
n-dimensional
mani olds
( o
he
de ini ion
o
he
na u al
bund-
le
see
[71
o [81)
.
Di e en iable
sec ions
o
he
bundle
F(M)
-->
M
a e
called
ields
o
geome icobjec s
.
Fde
assu-
me ha
F
is o
o de
.
I
means ha
i
U,V
a e
n-dimen-
_sional
mani olds
and
9,

:
U
-->
V
a e
di eomo phisms
such
ha

jx

=
jx
0
o
a ce ain

x
E
U

hen

F
(-p)
I
-l
17
(x)
=
F(O)

,
ha
is
F(9)
and
F(O)
a e
equal
on
he
ibe
o
7
:
F(U)
->
U
abo e
he
poin
x
.
Le
F
0
=
7
(0),
whe e

7
:F
(IR
n
)

_

IR
n
,

be
called
a
s anda d
ib e
o
he
bundle
F
.
I
X E
TM hen
he
low
o
X
induces
a
low
on
F(dom(X))
.
The
ec o
ield
on F(dom
(X))
de ined
by
his
low
is
called
a
comple
li
o
X
and
is
usually
deno edby
F (X)

(c
.
[9])
.
A
unc o
H
which
a aches
o each
n-dimensional
mani
old
M
he
p incipal
ib e
bundle
H
(M)
is
an example
o
a
na u albundle
.

I
W
:
N
i
->
N
2 is
an
embedding
o
n-dimen
sional
mani olds

hen

H
(,)
(j )
=
j0(~p
. )

whe e

j
EH
(Nl
) .
The
bundle
H
is o
cou se
o
o de
.
A
comple
li o
X
E
TM
o
he
bundle
H
(M)
willbe
deno ed
by
H
(X)
.
The
concep
o
'na u albundle'
was
in oducedby
A
.
Ni-
jenhuis
([7])
as
a mode n
app oach
o
he
classical
heo y
o
geome ic
objec s
(c
.
[l])
.
Le
a
be
a
iel
o
geome ic
objec s
on
M
.
Then
a
induces
a
mapping
a
:
H
(14)
-
F
0
such ha
a

0
)

=

F(
-1
)a (
(0))
.
Le

1
0
E
im
a
.

We assume
ha
hese

E
=

(
U
)
-1
(1
0
)

is
a
p incipal
subbundle
o
H
(M)
.
I
is
easy
o
see
ha
his
is
equi alen
o
he
ollowing
ac
:
o
each
x,y
E
M
he e
exis s

-p
EI'(M)

such
ha

F(sp)a
(x)
= a(y)
.

In
he
o he
wo ds
onecansay
ha
a
is
0-de o mable
(c
.
[10])
.
Conce -
ning
ields
o
geome ic
objec s
one
usuallyassumes ha hey
a e
0-de o mable
.
The
bundle
E
is
no
uniquely
.de e mi ed
by
a

and
i
also
depends
on
he
choice
o an
elemen
o

1
E
F
0
.
Al houghwe
ix he
bundle
E,
all
he
esul s
a e
ue
o
all
subbundles
o
H
(M)
induced
by
a
.
II
.
LIE
EQUATIONS
ASSOCIATED
WITH
THE
FIELDOF
GEOMETRIC
OBJECTS
.
We shall ecall
some
ac s
om
he
hoe yo
Lie
equa-
ions
.
These
ac s
will
be
applied
in
he
hi dpa
o
his
pa-
pe
.
The
basic
de ini ions
o
he
hoe y
o
Lie
equa ions
one
can
ind
in [21,
[31,
[4]
.
.
1
.
The e
a e
gi en
he
na u albundle
F
o
o de
and
he
ield
o
geome icobjec s
a
on
M
(c
.
I
.3)
.
Le
Ii
(M)

=

{j ,

:

~p
E
(M)

and

x
E
dom

p}
.
Then
n (M)
is
a
Lie
g oupoid
(c
.[6])
.
Le
now
n
(a)
=
{ix~P
E
n (M)
:
F(Sp)a(x)
=
a(~0(x))}
.
Wi h
he
ield
a
we associa ed
also
he
bundle
E
.
Lemma
II
.l
.
The
ollowingequali y
holds
II
(a)

=

{p
,
-P
-
1

:

P
.P'

E
E}
whe e
(j
g)
.
(j )
-1
=
j (g
.
-1
)
o
j
,j g
E H
(M)
and
x
=
(0)
.
P oo
.
Le
jx,
E'II
(a)
and
y =
W(x)
.
Le
j
E
Ex
hen

a
(j
(W° ))

=
F(('° )
-1
)
a
(Y)

=
F( -1)F(~p-1)a(Y)

_
=
F(
-1
)a(x)
_

so
j (cp° )
EE
.
Hence
we
ae
ha
,
,
= ~

-1
jx~

7
0
(
.P_ )
.
(jp
)

and
bo h
je s
on
he
igh
side
o his
equa ionbelong
o
E
.
Since
hen
j
W
E
{p'-p
_1
:
p',
p E
E}
.
Le now
j
=
j (h°g
-1
)
whe e
j
h,j g
E E
.
Then
F(O)a(x)

=
F(h°g
_1
)a(x)

=
F(h)F(g-1)a(g(0))

=
F(h)T
a
(j g)

_
=
F(h)
0
=
F(h)T
a
(j
h)

= F
(h)

F(h-1)a(h(0))

=
a(h(0))
.
Hence
we
ge
ha
F(O)a(x)
=
a(O(x))
.
because
i is
associa edwi h
he
bundle
E
.
A
local
di eomo phism
9
is
called
a
local
solu ion
o
he
non-linea
Lie
equa ion

II
(a)

i
o
each

x
E
dom9
'G
jx
E
II
(a)
;
II
(a)

is
calledcomple elyin eg able
i
o
each
i7EII
(a)
he eexis s
a
localsolu ion
9
such ha
j
W =

o
a
ce ain
xE
dom~p
.
x
2
.
Le now
R
(a)
_
{ixX
:
X
E
TM,
xE
dom
X
and
(LX0)
x
=0}
.
A
.
Zaj z
p o ed
ha
R
(a)
is
a
linea
Lie
equa ion
and
he
The e o e
by

[6)

we
ge
ha

1I
(a)

is
a
Lie
g oupoid
canonical
p ojec ion
p
:
R
(a)
-->
TM
is
su jec i e
whe e
p (j X)
= Xx
.
He e
Lx
a
deno es
a
Lie
de i a i e
o
a
ield
o
geome icobjec s
(c
.
[9],
[101)
.
A
ec o ield
X
E
TM
is
called
a
local
solu ion
o
R
(a)
i
o
each
x E
dom(X)
we ha e
j
XX
E
R
(a)
.
The
li-
nea
Lie
equa ion
R
(a)
is
called
comple elyin eg able
i
o
each

E
R
(a)
he e
exis s
a
local
solu ion
X
such
ha
jXX
=
n
o
a
ce ain
x
E
dom(X)
.
Lemma
11 .2
.
The
ollowing
condi ions
a e
eáui alen
:
i)
.

(L
x
a)
x=
0
;
ii)
.
H
(X)
z
ET
z
E
whe e
z
EEx
and
H
(X)
is
a
com-
ple e
li
o
he
ec o ield
X o
he
bundle
H
(M)
.
P oo
.
We ha e
he
ollowing
canonical
mapping
H
(M)
-
F(M),
whe e
4>(j
)
=
F( )l
0
o j
E
H
(M)
.
I
was
shown
ha
i
is
a
di e en iable
ib e
mapping
co e-
ing
he
iden i y
mapping
on
M
.

(c
.
[10])
.
I is
easy
o
see
ha
(D
1
(a(M))
= E
.
Hence
TE
=
(d4)-1(Ta(M))
.
Le
us
also
ema k
ha
i Z
is
a
ec o
ield
on
M
hen
H
(Z)
is
p ojec able
on
F(Z)
ia
Le
i s assume
ha

(L
x
a)
x =
0
.
I
means
ha
dx
a(X
x
)
=
F(X)a(x)
and
F(X)a(x)
E
TG(x)a(M)
.
We ha e
also
ha

d
z
(D(H
(X)
z
)

=
F(X) a(x)
.
Hence

H
(X)
E
T
z
E

and
.
he
impli
ca ion
i)
.
-
ii)
.
is
p o ed
.
Le now
assume
ha
H
(X)
z
E
T
Z
E
.
The e o e
d
z
ID(H
(X)
z
)

E
T
a
(x)
a(M)

and
mo eo e

F
(X)
a
(x)

ET
a
(x)
a(M)
.

Le
us
no ice
ha
he
canonical
p ojec ion
da(x)I
.
Ta
(x)
G
(M)
-

Tx
M

is
an
isomo phism
and

Xx =
dU(x)
(F (X) a(x)
)
d
a
(X)
T
(d
x
a
(X
x
)
)
.

Hence

F (X)
U
(X)

=
dx
a(X
x
)

and
his ends
he
p oo
o
he
implica ion
ii)
.
-
i)
.
III
.
HOMOGENITY
AND
TRANSITIVITY
OF
FIELDS
OF
GEOMETRIC
OBJECTS
.
In
hissec ionwe sugges
no ions
o
homogeni y
and
ansi i i y
o
ields
o
geome ic
objec s
.
We
also
do
his
in
he
in ini esimal
case
.
Then
we
ela e
his
no ions
o
he
simi-
la
p ope ies
o
E
.
1
.
We
shall
use
he
ollowing
no a ion
:
A
(a)

=

{
,
p

E

P
(M)

:

x E
dom9
-
F(W)a(x)

=
a(~p(x))}
(a)

=

{X
E
TM

:

SP
E A
(a)

whe e
~p
is
he
low
o

X}
.
De ini ion
III
.1
.
We shallcall
a
homogenous
i
A(a)
ac s
ansi i ely
on
M
and

in ini esimally
homogenous
i
o
e e y
x
E M
and
E
T
x
M
he e
exis s
XE
A(a)

such
ha
X =
.
x
De ini ion
111
.2
.
We shallcall
a
ansi i e
i
o
each
.
n E
II
(a
 he e
exis s
W
E
A(a)

such ha
j
w
x
whe e
x
is-asou ceo
77
.
We shallcall
a
in ini esimally
ansi i e
i o
each
1
E R
(a)
he e
exis s
X
E
A(a)
such
ha
j X
=

whe e
y
is
a
sou ce
o
The
se s
A(a)
and
A(a)
can
be
desc ibed
in
he
ollo
wig
.
way
:
Lemma
111
.3
.
I
a
is
a
0-de o mable ieldo
geome-
ic
objec s hen
a)
.
A(a)
is
a
se
o
solu ions
o
II
(a)
;
b)
.
A(a)
is
a
se
o solu ions
o
R
(a)
.
P oo
:
I
9
E
F(M)
hen
T
is
a
localsolu ion
o
II
(a)
i
o
each
xE
dom~p
j ,
E
II
(a)
.
This
is
equi alen
o
he
ac
ha
o
each
xE
domsp
F(W)a(x)
=
a«p(x))
.
Hence
;p

is
a
local
solu ion
o

II
(a)

i

p
EA
(a)

So

a)
.

is
p o ed
.
Le
XE
A(a)
and
x E
domX
and
le
sp
deno e
he
low
o

X

hen

F
(sp
)
a
(x)

=
a
0p
(x»
.

Hence

F
(X)
a
(x)
dx
a(X
x
) .
This
means
ha
(LX
u)
x =
0
because
he
Lie
de i a i
e o
he
ield
o
geome ic
objec s
can
be
exp essed
in
he
ollowing
way

(L
X
a)
x = dx
a (Xx
)

-

F
(X)
a
(x)

(c
.

[91,[101)
.
The e o eX
is
a
solu ion
o
R
(a)
.
Le now
Y
be
a
solu ion
o
R
(a)
hen
i
gene a es
he
ield
F(Y)
on
F(M)
.
Since
o
e e y
xE
domY
F(Y)a(x)
=
d
x
a(Y
x
)

hen

F(Y)
la(domY)

is
a
ec o ieldon

a(domY)
.
I
p
is
a
low
o
Y
hen
F(0
)a(x)
E
a(domY)
.
Hence
F(Ví
)a(x)

=
a(41
(x))

and

Y
E
A(a)
.
.

This
ends
he
p oo
o
b)
.
F omlemma
111
.3
we
ge
immedia ly
he
ollowing
co o-
1
lla y
Co olla y
111
.4
.
i)
.
The
ield
a
is
homogenous
i
o
each
x,y
E
M
he eexis s

~p

a
local
solu ion
o

I
(a)

such ha

~p
(x)
=y
;
ii)
.

a
is
in ini esimally
homogenous
i
o
each
x
E M
and
E
TxM
he eexis s
X
a
local
solu ion
o
R
(a)
such ha
Xx
=
;
iii)
.
a
is
ansi i e
i
11
(u)
is
comple ely
in e-
g able
;
i )
.

a
is
in ini esimally
ansi i e
i
R
(a)
is
comple ely
in eg able
.
2
.
Wi h
he
p incipal ib ebundle
E
we
can
associa e
he
ollowing
se s
:
A
(E)

_

{Sp

E

P

H
(
~0
)
E
Idomp

C

E
}
A(E)

_

{X
E
TM

:

p
EA(E)

whe e
`p
is
he
low
o X}
The
bundle

E

is
calledhomogenous
i
o
each

x,y
EM
he e
exis s

E
A(E)

such
ha

q
(x)

=
y
;

E

is
called
an
si i e
i
o
each
zl,z
2
E E
he e
exis s
0
E
A(E)

such
ha
H
(,,
,
)z
l
=
z2
.
The
bundle
E
is
called
in ini esimally
homogenous
i
o
each
xE
M
and
each
ETx
M
he e
exis s
XE
A(E)
such
ha
Xx =
;
E
is
called
in ini esimally
an
si i e
i
o
.
each

z
E
E

and

each

X
E T
Z
E

he e
exis s
X
E
A(E)

such ha
H
(X)
z =
X
.
Suchmeanings
o
(in ini esi
mal)
homogeni y
and
(in ini esimal)
ansi i i y
a e
used
o
ins ance
by
P
.
Molino
[51
.
Lemma
111
.5
.
I
E
is
a
p incipal
ib ebundle
associa
ed
wi h
he
0-de o mable ield
o
geome ic
objec s
a
hen
A(a)
=
A(E)
and
A(a)
=
A(E)
.
P oo
.
I
is
enough
o
p o e
ha
A(a)
=
A(E)
.
Le
W
E
A(a)

hen
i
is
easy
o
no ice
ha
a
.H OP)
_
-
Ta1H (dom-p)'

I

z
E
EldomW

hen

T
.(H
(SP)z)
='
,
a
(z)
=
lo
-
Hence

H
«p)
El
dom(p
C
E

and

-p
E A
(E)
.
On
he
o he
hand
le
0
E
A(E)

and
le
j
É
Ex
whe-
e
xE
domo
.
I
implies
ha
a
(H
(0)j' )
=
a
(j
)
=
0
.
F om
he
de ini ion
o

a
we
ge
ha

F(
-1
.,
y-1
)a0(x)

=
=
F( -1)a(x)
.

Hence

F

(Vi
-
1
)aO(x)

=
a(x)

and

,~
E
A(a)
.

Tha
ends
he
p oo
.
In
he
ollowing
p oposi ion
we
compa e
(in ini esimal)
homogeni y
and
(in ini esimal)
ansi i i y
o
he
bundle
E
and
he
ield
o
geome icobjec s
a
.
P oposi ion
111
.6
.
I
a
is
a
0-de o mable
ield
o
geome ic
objec s
and
E
is
a p incipal
ib ebundle
gene a ed
by
a
hen
i)

a is
homogenous
i
E
is
homogenous
;
ii)

a is
in ini esimally
homogenous
i
E
is
in ini
esimally
homogenous
;
iii)
ais
ansi i e
i
E
is
ansi i e
;
i )

a is
in ini esimally
ansi i e
i
E
is
in ini
esimally
ansi i e
.
P oo
.
The
i s condi ion
is
a
simple
consec
;uence
o
Lemma
III
.5
.
;
ii)
.
one
can
easy
ge
omlemma
11
.2
.
Simila y
i )
.
easly ollows
om
lemma
11
.2
.
and
lemma
111
.5
.
To
p o e

iii)
.
i is
enough
o
no ice
ha
i

z
1
,z 2
EE
and

WE
I'
(M)

hen

H
(SP)
zl
=
z2
i

jx~p
=

z2
.
zi1

whe e

x