521
Internat. J. Math. & Math. Sci.
Vol. 8 No. 3 (1985) 521-536
LOCALLY CONFORMAL SYMPLECTIC MANIFOLDS
IZU VAISMAN
Department of Mathematics
University of Halfa, lsrael
(Received April 4, 1984)
M2n(n
M
,
.
(M2n,fl)
where
A locally conformal symplectic (l.c.s.) manifold is a pair
on
2-form
nondegenerate
a
> i) is a connected differentiable manifold, and
k9 U s (U s- open subsets)
/U e
o U s -IR, d O.
such that M
ABSTRACT.
Equivalently,
d
^
.q
for some closed 1-form
L.c.s. manifolds can be seen
as generalized phase spaces of Hamiltonian dynamical systems since the form of the
Hamilton equations is, in fact, preserved by homothetlc canonical transformations.
The paper discusses first Hamiltonian vector fields, and infinitesimal automorphisms
0, we say
(i.a.) on l.c.s, manifolds. If (M,) has an i.a. X such that (X)
e
de
assumes the particular form
Such an M is a 2-contact manifold with the structure forms (,e), and it has a
vertical 2-dlmensional foliation V. If V is regular, we can give a flbration theorem
bundle over a symplectlc manifold. Particularly,
which shows that M is a
that
M
^
is of the first kind and
T2-principal
9
is regular for some homogeneous l.c.s, manifolds, and this leads to a general con-
struction of compact homogeneous l.c.s, manifolds.
Various related geometric results,
including reductivity theorems for Lie algebras of i.a. are also given. Most of the
proofs are adaptations of corresponding proofs in symplectlc and contact geometry. The
paper ends with an Appendix which states an analogous fibratlon theorem in Riemannlan
geometry.
s-contact manifold,
KEY WORDS AND PHRASES. Locally conformal symplectic manifold,
Boothby-Wang fibration.
58F05.
7980 MATHEMATICS SUBJECT CLASSIFICATION CODE. 53C15,
i.
INTRODUCTION.
A symplectic manifold is a pair
(M2n,),
where
M
2n
is an even dimensional dif-
and connected) and fl is a
ferentiable manifold (all our manifolds are assumed C
closed nondegenerate 2-form on M. Such manifolds are very important since they provide a good geometric framework for Hamiltonian mechanics, and for other chapters of
theoretical physics.
If the 2-form
is nondegenerate but not closed,
(M2n,fl)
is
an almost symplectic manifold, and this definition provides a class of geometrically
I. VAISMAN
522
In between, the locally conformal symplectlc (l.c.s.) mani2n
(n > I) which have an open
M
interesting manifolds.
folds are defined as almost symplectic manifolds
{Us}
covering
Equivalently,
o
and a system of functions
6 A’
do s glue up to a closed 1-form
d
O.
d
Clearly,
) =0.
d(e
such that
(i.i)
A
Formula (i.i) was established by H.C. Lee [i
well defined, and
Uand
the Lee form;
and we call
is exact, the manifold is
iff
it is
globally con-
formal symplectic (g.c.s.).
for the first properties and
and [5
We refer the reader to [2 ], [3 ], [4
examples of l.c.s.manifolds,that also provide a geometric motivation for the study of
this class of manlfolds. But, let us also point out a physics motivation. Indeed, let us
look at a dynamical system with
M
Us
with the local coordinates
and the dynamics consists of the
Every point of
orbits of a well defined vector field X
hood
Then its phase space can be
degrees of freedom.
n
seen as a 2n-dimensional differentiable manifold
(qi(s) Pj"a))
and momenta, and there is a Hamiltonian function
(i
M
has an open neighbour-
n)
j=l
given by positions
H(a) (qi(a), P j())
such that the
orbits are defined by the Hamilton equations
i
dq(e)
(c)
H(s
dPi
H(s)
dt
8q(e)
s)
dt
Pi
6
The well known symplectic interpretation
Hamiltonian field of
(1.2)
i
tells us that
X
with respect to the symplectic form
H()
is precisely the
(e)
(=)
i=l
s)
Now the usual continuation of this interpretation consists in asking the local
(s)
forms
and local functions
X
to glue up to a global symplectic form
H(e)
and
But this is not compulsory since the only global entity is
H.
a global Hamiltonian
and we must only ask that the transition functions
i
q(B)
q(8)(qcs) Ph(e)), Pi
(8)
k
i
preserve the form of the Hamilton equations (1.2).
implies
8
8 IBsa E8e
if we take
H(8
(8)
k
Pi (q(a)
p
h()
(1.3)
This happens not only if (1.3)
(i.e.,(l.3) are canonical transformations), but also if (1.3) implies
const.# 0 (i.e., (1.3) are homothetic canonical transformations)
In other words, we get the Hamiltonian dynamics if the
18sH(a)
geometric structure of the phase space is defined by an open covering
{Us}a
a corresponding system of local symplectic forms
UUB#
has
8 18
18s
such that over
6 A
and
one
(1.4)
const.
In this case, we get easily from (1.4) the cocycle condition
B7 B
hence we have a basic line bundle
L
on
L
we have a Hamiltonian cross-section of
(1.5)
M, and instead of a Hamiltonian function
(a "twisted Hamiltonian").
It is well known that the cocycle condition (1.5) implies
l
for some functions
o Ue+
IR
e
/
(1.6)
e
defined up to a te
f/ue
(f:M +), and then
(1.4)
523
LOCALLY CONFORMAL SYMPLECTIC MANIFOLDS
shows that
e
(1.7
M
2-form on
global non-degenerate
is a
a
Hence (M,)
defined up to a global factor.
is an l.c.s, manifold.
Therefore, the l.c.s, manifolds are natural phase spaces of Hamiltonlan dynamical
systems, more general then the symplectic manifolds, and this is the announced motivation.
.
Finally, we indicate that the l.c.s, manifolds play an important role in the recent works of A. Lichnerowicz
].
In this paper, we do not intend to discuss problems of mechanics or physics, but
some problems concerning the differential geometrical structure of the l.c.s, manifolds.
.
In Bectlon 2, we discuss infinitesimal automorphisms (i.a. of an l.c.s.
structure
X
If there is an i.a.
called of the first kind, and
manifold [12]
such that
(X)
O,
the manifold
has a particular form, while
(M,)
This happens necessarily if
M
Boothby-Wang
We also deduce that a homogeneous l.c.s, manifold
9
8
is
is homogeneous nonsymplectlc. In
Section 3,we define regular l.c.s, manifolds, and give a corresponding
fibration theorem
(M,)
becomes a 2-contact
with an invariant i.a.X such that
(X) # 0
is regular.
In Section 4, we discuss
compact homogeneous i.c.s, manifolds, and show a method for constructing such manl2
folds as T (torus)-bundles over compact homogeneous simply connected Hodge manifolds
by applying the results of
I0] for contact manifolds.
We also apply the method of
[I] in order to derive some reductivity results for Lie algebras of i.a. of l.c.s.
structures
In each section we also give various other related results.
Most of the
proofs are adaptations of corresponding proofs in symplectic and contact geometry.
The paper closes with an Appendix where we give a Riemannlan analogon of the BoothbyWang fibration theorem.
This text is a part of a series of lectures on Boothby-Wang fibration theorems
given by the author at the Istituto Matematico del Politecnico dl Torino (Italy),
under the invitation of the Italian Consigllo Nazlonale delle Rlcerche (C.N.R.). I
should like to express here my thanks to the CNR of Italy and to my hosts in Torlno,
particularly prof. F. Tricerri.
2. HAMILTONIAN FIELDS. INFINITESIMAL AUTOMORPHISMS.
Let
(M2n,)
,
be an l.c.s, manifold with the Lee form
such that (i.i) holds
Then, we also have the characteristic vector field A defined by
(and
d
0).
i(A)
--,
and it is easy to get
i(A)
O,
LAW
LAn
O,
C (M) denote the associative
one such function. As in Section i,
has
well defined associated cross
e
Then, the usual symlectc
local fields
given by
Let
0
(2.1)
C-fuctlons on M, and f M-IR be
there is a well defined line bundle L on M, and f
section
of L iven by the local functions f
HamItonian formalism [6] provides us with the
algebra of
fL
Xf
i(Xf)
But (2.2) is equivalent to
i(Xf)
glue up to a global vector field
df
df
Xf
i(Xf)
(2.2)
f
which shows that the local fields
Xf
defined by
df
f
(2.3)
I. VAISMAN
524
will be called the Hamiltonian vector field of f with respect to the 1.c.s.
defines the dynamics of the local Hamiltonians fa as de-Clearly,
form
Xf
Xf
scribed in Section I.
Using these fields, we define now the Poisson bracket
{f,g
PM)=(C(M),
The last expression of (2.4) shows that
M)
vector fields of
H
(2.5)
is the Lie algebra of the
x(M)
(where
x(M)
P(M)
is a Lie algebra homomorphism.
Xf
f
given by
is a Lie algebra (called
(. ,.})
[Xf,Xg]
X{f,g}
or, equivalently, the mapping
(2.4)
and that one has
(M,))
the Poisson-Lie algebra of
ea{f,g}
-Xgf+f(Xg)=
Xfg-g(Xf)
(Xg,Xf)
The following fact is rather interesting.
Tlen
H
(M,) be
Let
PROPOSITION 2.1
(connected’.)
a
g.c.s.
l.c.s, manifold that is .not
is a monomorphism.
By (2.3),
PROOF.
0
Xf
0
df -fm
means
cannot be
f
and
nowhere zero
0 for
M would be g.c.s. Hence, let f(x o)
(d/) ( # 0) on some open connected neighbourhood of x o
0 on that neighconst., and f
O, whence f
gives d(f)
since otherwise m would be exact, and
Xo 6 M
and put
0
df -fm
Then
bourhood.
neighbourhoods
and
x
m/U.1
Un
U I,
such that
n).
is exact (i =i,
REMARK.
0
Xf
Therefore,
to x
one can build a chain of open connected
x 6 M
Now, for an arbitrary
Xo
x 6 U
I
U
n
Ui+I
i
n-l),
# (i=l,
propagates along this chain from
0
f
Then
0.
f
implies
6 U
This result is not true on g.c.s, manifolds.
Furthermore, it follows from (2.3) that any Hamiltonian field satisfies
Lxf
(2.6)
(Xf)
is not an infinitesimal automorhence, generally and unlike in the symplectic case,
by
O, and form a
defined
are
latter
the
phism (i.a.) of (M,). Of course,
Xf
bracket-Lie algebra
of (2.1) and
x(M).
(2.3), Af
We do have
Xf 6 x(M)
0. Vector fields
X
iff
m(Xf)
such that
0 or, equivalently in view
re(X)
0 will be called
horizontal fields.
Now, let us refer
Lx=O
then
,
to an arbitrary
X 6
x(M).
and, by (i.I),
x(M),
m(X)=const., m(Y)=const., and dm(X,Y)=O yield m([X,Y])=O. Hence, the application
x(M) ]R defined by (X)=m(X) is a Lie algebra homomorphism for the commutative
Lie algebra structure of
JR. We call
the Lee homomorphism of
is the Lie algebra of the horizontal elements of
X 6
LX=-O
Then we have
as well. The later condition implies m(X)=const. Particularly, if X,Y 6
x(M)
x(M),
with (X)#O will be called transversal i.a.
the l.c.s, manifold M is of
the first kind if it has
second kind, and the Lee homomorphism is trivial.
x(M).
denoted
The kernel ker
xhr(M).
The i.a.
(t.i.a.), and we shall say that
t.l.a. Otherwise, M is of the
If m has vanishing points, M
necessarily of the second kind. Hence, if M is of the first kind m
is
0 everywhere, and,
(M,) is of the
has the Lee
), then (M,e
if M is compact, M has a vanishing Euler-Poincar characteristic. If
first kind, and f:M ->jR is a function such that df
/Xo--(x
-fn
form 0-df with a vanishing point, and it is %... of the second kind. Clearly, if M is
525
LOCALLY CONFORMAL SYMPLECTIC MANIFOLDS
sequence of Lie algebras
is onto, and we have the following exact
c:
hor
of the first kind
0
-+ IR
Xp,(M)
(M)
X
(2 7)
O.
about the l.c.s..anifolds
It turns out tha we c. obtain much more information
and call B the
B E
of the first kind. Indeed, let us fix an element
has a unique decomposition
basic t.i.a, of (M,f). Then, every Y
0-i(i) = f(M),
(M)
X
Y
+ (Y)B, X E
hor
Xf
(2 8)
(M)
(2.9)
co A 0
dO
fl
0
0 as i(B)dfl 4- di(B)fl
LBf
0), and write down
-i(B) fl (hence 0(B)
for f namely
expression
This yields a particular
Now, put 0
Furthermore, we have
-i(B)(f-
-i (B) di (B) f
0 =-LBi(B)fl
(2. lO)
0
i(B)df)
’hence
fin4
But then (2.9) and
and rank dO < 2n
n-1
(d0)
^0^
(2.11)
0
i(B)dO
0
4
yield
(2.12)
0
This yields
everywhere.
2n
admits an l.c.s, structure of the first kind
A manifold M
O, rank dO < 2n, and (2.12) holds at
iff it admits two l-fvrms ,0 such that d
every point of M.
PROOF. Above, we obtained ,0 from fi
Conversely, if ,0 are given,
PROPOSITION 2.2.
(2.9) yields an
(B)
i
define a unique vector field
LB
0
LB0
Hence
0
Of course,
(,0).
with Lee form
.
0(B)
0
0
B
M
l.c.s, structure
and
0
B
on
i(B)d0
(2.13)
(that also satisfies
uniquely, but
f
-0)
i(B)
such that
.E.D.
is a (basic) t.i.a.
define
Then the equations
does not define uniquely
Note also that
(A)
0(A)
0
I
i(A)d0
define the characteristic vector field of
it also preserves
A.
This means
M (i(A)
[B,A]
O,
(2.14)
0
),and
since exptB) preserves
M
and we obtain on
the vertical
2
V
foliation
whose leaves are the orbits of a natural action of
span {A,B}
In the next Section, we shall use V in order to get more geometric information on M.
In connection with the above discussion, we shall also make the following complementary considerations. Formula (2.6) proves that a Hamiltonian field is a conformal
infinitesimal transformation(c.i.t.)of (M,). Generally, a vector field X of M
is a c.i.t, if
[Lf]
LX aXf
where
by
X
x(M),
is a function on
M.
The c.i.t, form a bracket Lie algebra to be denoted
and if besides (2.15) one also has
L[x,y]
(2.15)
Lyf
ayfi
it follows
(Xay- Yax)
The Hamiltonian fields form a Lie subalgebra
(2.16)
XHam(M)
of
c
(M).
Now, if
X
saris-
I. VAISMAN
526
fies (2.15) then, by differentiating this condition and since
LX
n > 1
we get
(2.17)
dx,
whence it follows that
(X) + k
aX
If this
(2.18)
const.
are the local symplectic forms of the l.c.s, structure.
where
iff it is an infinitesimal homothety of the forms
c.i t
re(X)
(X)
([X,Y])
-k(k of (2.18)).
aX
-d(X,Y)
If
X,Y 6
(M),
e It
such that
X
is a
C
x(M)
and we have
/IR
given by
(2.16),
it follows
is also a Lie algebra homomorphism.
hence the extended
0,
Its kernel consists of fields
fields, and we denote ker
k
X
Hence
C
we can extend the Lee homomorphism to
Furthermore,
LX
we see that (2.15) is equivalent to
(2.15)
is used in
X
k
i.e., of locally Hamiltonian
0
XHam(M)XHam(M).
is precisely the locally Hamil-
tonian fields that should be interesting in mechanics.
ly
P.awiltonian
and we shall say
e
x(M), we have the
c
c
_+
0
XHam (M) x(M) IR
that -M has many c.i.t.
field in
for every
Y
-i(C)
y
Then, if
(C) +
^
X
y
X
(i), called a basic field which
XHam (M)
LC
(2 20)
i(C)d +di(C)=(C)fl+
^
y
if
LX X
we get
Indeed, if
LX X
with
X
Whence we obtain
(x)
=x-- (x) =x
(x)
-
C
(2.21)
(x)
on
(M,)
(C)
such that
I, and
-+efl
X
Hence the existence of many c.l.t, is a conformally invariant property.
that there is a vector field
i.e.,
is
More precisely, let us note that the
of (2.19) is conformally invariant.
and
aC
see that an l.c.s, manifold with many c.i.t,
a candidate of a manifold of the first kind.
+ dq0
dy
^
or equivalently
Hence, by comparing with (2.9), we
Lee homomorphism
-i
x(M)
+ (Y)C,
dy
we get
(2.19)
0
C
Y 6
we have
0
dy
following exact sequence of Lie algebras
C 6
If this happens, let us fix an element
gives uniquely
i.e., if there is a non-local-
is nontrivial
If the extended Lee homomorphism
then
+ Xp
Now, assume
d[ei(C)]
is a
degenerate 2-form for some function 0 on M. Then Proposition 2.2 shows that
is an l.c.s, manifold of the first kind.
(M,e)
(M,)
Let
hor
(M,B)
be an l.c.s, manifold of the first kind, and B a basic t.l.a. Let
hor
(M) whose automorphisms also preserve B
i.e.,
be the Lie subalgebra of
hor
X 6
(M,B) iff (X)
C(M)
the subset of
X
X
0,Lx
X
0,[X,B]
O.
On the other hand denote by
C (M) that consists of functions that are foliate with respect
O, and remember the ap0, Bf
or, equivalently, satisfy Af
plication H(f)
Then, we can prove
PROPOSITION 2.3. Let M be an l.c.s, manifold of the first kind which is not
to the foliation
Xf
g.c.s.
Then
C(M)
is a Poisson-Lie subalgebra
P
of
P(M),
and
H
sends iso-
527
LOCALLY CONFORMAL SYMPLECTIC MANIFOLDS
PV
morphically
Xf E xr(M).
O,
Af
since
f E
Let
PROOF.
hot
XR (M,B).
CV(M), and
onto
(2.22) reduces
Now
of
and since
B
it is injective because of
f
-8(X)
Bf
8(Xf)
0
(2.22)
df- f
implies
(2.23)
-f
LXf8
C(M)
to
O, and because
(M,B), and
hor
Xfl
(2.23).
O,
re(X)
(M,B) (which implies
O), and define
LX8
Then
i(X)
i(X)(d8
X
implies
Af
0
(X) 8 + 8(X)
df
f
Furthermore, as in the remark following (2.6),
Xf
dS(B,X)
(X) 8 + 8(X)
8)= i(X)d0
^
d(8(X))
LX8
i.e.,
-8(Xf)
df, which with (2.23) implies
[Xf,B] O. Hence H sends
hor
X
Conversely, let
+
i(Xf)d8
to
we also have
0
Lxffl
By (2.6) and the remark afterwards,
Xf.
Then, by (2.3) and (2.9)we get
i(Xf)d8
which applied to
H(f)
O, and we also have (8)(B)
X(8(B))
8([B,X])
B(8(X))
B(8(X)),
(Xf)
(X)
0.
Bf
i.e.,
0
-8([X,B])
Hence H
8([X,B])
X(8(B))
0
is
also a surjection for the sets of Proposition 2.3.
(M,B)
X{f,g}=X
{f,g}
Cv(M ).
C (M)
f g
Finally, let
hor
and
hor
Xf Xg Xfl
therefore
H
and, since by Proposition 2.1
(M,B).
(M,)
_
is homogeneous if it admits a transitive Lie group
An
PROPOSITION 2.4.
of
-preservlng
Let
(M,) be
a homogeneous l.c.s, manifold which is not sym-
Then it is necessarily a manifold of the first kind.
plectic.
PROOF.
may
G
l.c.s, mani-
(In Section 4, we shall give a rather general construction of such
manifolds.
# 0
g
_O..E.D.
diffeomorphisms.
G
X
is inJective, we must have
We close this Section by another simple but interesting result.
fold
[Xf
Then
be
assumed connected as well.
M
everywhere, and
leaves of this foliation, and let
M
Since
is foliated by
y E G
X (a
G.
These elements have associated vector fields
XlO...oex
for some elements
for at least one index
p
to
COROLLARY 2.5.
q.
e
Let
such that
exp
p
is not symplectic and homogeneous,
0
m
y
cannot send
Then the homogeneity .roup
Remember that all our manifolds are connected.
since otherwise
be points on different
p,q
y(p)
q.
k)
1
Then we may write
of the Lie algebra
X x(M),
and we must have
(Xa)O
O, and
acts along the leaves of
y
of
it
Q.E..
A semisimple Lie group
G
cannot act transitively on a nonsym-
plectic l.c.s, manifold.
PROOF.
algebra g’
Indeed, if G is semisimple its Lie algebra g is equal to the derived
But we know that any bracket of i.e. is horizontal. Hence g would
consist only of horizontal fields, which contradicts Proposition 2.4.
3.
REGULAR L.C.S. MANIFOLDS.
In Section 2, we saw that an l.c.s, manifold of the first kind has important
foliations
Inspired by a corresponding theory of contact manifolds
define the regular l.c.s, manifolds as l.c.s, manfolds
M
8
],
we shall
of the first kind for which
528
LOCALLY CONFORMAL SYMPLECTIC MANIFOLDS
it is possible to choose a basic t.i.a. B such
that the corresponding vertical foli]
span t’A,B
is simple (regular) and the
corresponding space of leaves M/];N
is ,] Ilausdorff differentiable
manifold. Under these hypotheses, we may expect a fibration theorem, and, in fact, such a tl’orem
was given for a more general structure
ation
[12], [9
Namely, let
M
2n+s
1
consists of s 1-forms
be a differentiable manifold. An s-contact
structure on M
s
and one 2-form q of rank 2n such that
i
s
A
# 0 (everywhere)
A
n
^
uf
dmU
(a
(3.11
u
s), d
const." u=l
0
u
const, is a consequence of the other relations.
n > 1, a
If at least one of
u
the
is nonzero, the structure will be called of nonzero type.
s-contact
If
I/ where
structure defines a decomposition TN
C
s
given by
0}
0
m
0, and /-- {X/i(X)f
C
ml
over, we have
s
(the horizontai bundle) is
(the vertical bundle). Note-
uniquely defined basic vertical vector fields
E (u
1,..., s)
u
given by
aUv
i(Ev)mU
I.,
0 (u,v
i(Ev)
s)
(3.2)
This relations imply
v
mu i(Ev)dmU
e
LEv
i([E
u
i(E )a
(3.4)
0
i(Ev)LEut
Ev])t LEui(Ev)t
i([Eu,E v
])
L
(3 3)
0
v
E
i(E )-
i(Ev)LEu
v
u
0
(3.5)
0
whence
[E ,E
u v
Hence
0 (u,v,
M
is a foliation of
this foliation is simple, and
s)
1
by the orbits of a natural action of
N
M/V
is
Hausdorff, we say that
s-contact manifold, and the following result holds
PROPOSITION 3.1.
manifold, and p:
T
s
Let
M2n+s N2n
(3.6)
(M2n+s
m
u
,
[12],
M
s
is a
M.
on
If
r
9
be a compact connected regular s-contact
the corresponding submersion.
Then
p
is a principal
’
(torus)-bundle, N is a symplectic manifold with the symplectic form
such
u
p*’
and there are some constants c
such that {cUreu} is a connection
that
p
on
with the curvature proportional to
PROOF.
’
The regularity hypothesis implies the regularity of the foliations by the
orbits of E
for each u=l
u
Then, for each
is a constant
u=l,...,s,
Cu
# O,
,s,
whence these orbits must be embedded circles [8 ].
inf{t/t > O, exp(tEu)(X)=X}
Eu(X)
theorem [13 applied to the pair
the period function
in view of
Tanno’s
(Eu,mU).
Now,
we see that the bracket con:nuting vector fields
yield a free right acu
s
tion of T
on M. Furthermore, let U be a cubical flat regular coordin-
(i/cu)E
s/ m.s
ate neighbourhood for (M,V) with the coordinates
a
const, on the leaves of
such that x
Put
P
-1
where
(u’)
o
bYh((xa),
U’
M
(t
I
ts))
is given by
exp(
o(x a)
tl
c
E1)...
I
(x a,0).
(xa,xu)
(a
i
,2n; u--i
p(U), and define h:U
U’
exp(
Then
ts Es)((xa)),
s
c
h
,s)
T
s
(3.7)
is a diffeomorphism which
LOCALLY CONFORMAL SYMPLECTIC MANIFOLDS
gives a local trivialization for
T
such that the right action of
p
TS-component
tiplication on the
529
s
is right mul-
This proves the principal bundle struc-
in (3.7).
All the other assertions of Proposition 3.1 are clean from (3.2), (3.3),
ture of
p.
(3.4).
Q.E.D.
The following converse result is also clear.
(N
Let
PROPOSITION 3 2
2n
’) be a symplectic manifold
s
(u)
T -principal bundle endowed with a connection
const.)
(mu, p*,)
Then
is a regular
such that
and
p: M
u
u
d
s-contact structure on
N2n
2n+s
a
u
*
M.
Now, by Proposition 2.2, an l.c.s, manifold M admits associated 2-contact
2
i
O, a
i, and conversely. Moreover, if M
structures (m,0, dO) such that a
is regular the associated 2-contact structure can be assumed regular, and we have
M
Let
PROPOSITION 3 3
2n
be a compact regular I c s
manifold, and p:
M2n-N 2n-2
be the corresponding submersion on the space of the leaves of a regular vertical foliation
V
fold
N
M.
of
connection
M
then
Then
p
(m,0)
T2-principal
is a
Conversely, if
fibre bundle over the symplectic mani-
is such a principal bundle, and it is endowed with a
p
O,
d
such that
and
N,
projects to the symplectic form of
dO
is a regular l.c.s, manifold.
In
Proposition 3.3 provides a construction method for regular l.c.s, manifolds.
fact, it is easy to understand that
fibrations: first, we can project
p
M
can be obtained as a composition of principal
onto the manifold
P
B
of the orbits of
project
P
N
onto
by the Boothby-Wang fibration
Particularly, the symplectic form of
circle bundle.
8
cohomology class
8
which is again a principal
N
Conversely, the construction of
P
two steps: construct
like in
8
and then
M
and
Then
this will be a flat principal circle bundle over a regular contact manifold.
must represent an integral
M
will be realized in these
as a flat principal circle bundle
P
over
The results above are a straightforward generalization of the Boothby-Wang fibra-
8
tion theorem
7
Moreover, many of the other results of the basic paper
can
also be generalized straightforwardly to the present situation, and we shall indicate
here this generalization
PROPOSITION 3.4.
of the automorphisms of
PROOF.
U
Let
M
Let
M
acts transitively on
8
M.
(M,V)
be a cubical flat regular coordinate neighbourhood of
like in the proof of Proposition 3.1.
we see like in
Then the group
be a regular compact l.c.s, manifold.
that
G
Let
G
be the automorphism group of
acts transitively along the slices of
m
0
M.
in
Then,
U
But it also acts transitively on slices by the translations of the corresponding parameter.
Hence
G
acts transitively on
U, and, because of connectedness,
it also acts
transitively on M
.E.D.
In 3 it is shown that, if M is a compact connected l.c.s., and m # 0 everywhere, then the group of its conformal transformations acts transitively on M.
Furthermore, an s-contact manifold (M,
) is called homogeneous if M
G/K,
where G is a Lie group of s-contact automorphisms which acts transitively and
effectively on M, and K is a closed subgroup of G
Hence, V g E G, g u m u
g
(the second relation follows from the first in the nonzero
type case). Then
mu,
,
530
I. VAI SMAN
one has
PROOF.
The forms
8
which are ad K-invariant, and we shall look at the closed subgroup
G
on
m
H c G (H m K)
defined by
{h 6 G/(adh)
H
Then, if
X
K
be a homogeneous s-contact manifold of
)
is a regular s-contact manifold.
u
lift to corresponding left-invariant forms
m
M
Then
the nonzero type.
mu,
G/K,
(M
Let
PROPOSITION 3.5.
G,
denotes the Lie algebra of
(e
Te G
a generical element of
G,
ponding left invariant field of
{X E
h
H
of
}
k
and
G)
is the unit of
K, if we denote by
of
and
by
X
the corres-
we get:
LXu
/
j
(adh)
m
LX
0
O}
Since everything in the above construction is left-invariant, if we use the definition
X E h
of the Lie derivative, we see that the conditions which define
to
(dU)e(X,Y)__
[Y_, _Z])
Y, Z are arbitrary elements of g, and
where
for some index
L
u,
0
tion is the first condition
h
e(X--’
O,
{X
g /
e(X,Y)
0}
(3.8)
0
we also used
u
Lx
follows from
(3.8), which
0
2n
8
0
# 0
a
Since
fle(X’Y)
],
I.e.,
0
dim k
we get dim h
H m K
G
Furthermore, it is known that for a triple
d
and the only remaining condi-
is equivalent to
and since rank
are equivalent
+
s.
as above there is a na-
tural diagram of locally trivial fibrations
G/K
G
o
G/H
(3.9)
p
where, particularly, 0 has the structure group H
H/K. Now, if Eu are the basic vertical fields on
E
left-invariant lifts
G/K,
there are (non-unique)
0
i(Eu)
which satisfy
s-dimensional fibre
so that
E (e)
are in
u
It is easy to deduce from this that the tangent distribution of the leaves of
h
on
u
G
to
and the
G/K
is precisely the vertical distribution of the fibration
applying the Corollary on p. 28 of [14]
and its basis is a covering manifold of
REMARKS.
(G/K)/g
i) Just like in
G/(H0. K), where
is
8
G/H
is the connected component of
au
same argument as for Corollary 2.5 shows that, if
u,
G
V
is simple
i.e., a Hausdorff manifold.
we can see that the space of the leaves
H
0
proof of Proposition 3.5 also holds for all
index
But then, by
p
it follows that the foliation
0
G
if
a
u
0
e
in
H.
2)
The
is semisimple but the
holds for at least one
cannot be semisimple.
COROLLARY 3.6.
invariant t.i.a.
Let (M,) be a homogeneous l.c.s, manifold which admits an
M is a regular l.c.s, manifold.
Then
The supplementary hypothesis is necessary in order to have a homogeneous associated 2-contact structure, and to apply to it Proposition 3.5. We notice that if (M,) is
G/K, where G is a
a homogeneous nonsymplectic l.c.s, manifold such that M
group), then M must have an inthe proof of Proposition 2.4, there is an element X
reductive Lie group (particularly,
variant t.i.a.
Indeed, as in
G
is a compact
LOCALLY CONFORMAL SYMPLECTIC MANIFOLDS
But then (2.7) yields an exact sequence of Lie algebras
which generates a t.i.a.
hor
g
0
hor
where
__c
--+m
hor
(3.10)
0
M
which yield horizontal fields on
consists of elements of
is reductive
since
531
has an ad
reductive (compact) we shall say that
.E.P.
G-invariant complement.
M
and
G
If
is
(compact) type.
is of the reductive
COMPACT HOMOGENEOUS L.C.S. MANIFOLDS.
4.
Like for the regularity property, we may expect to obtain information about corn-
pact homogeneous l.c.s, manifolds from a discussion of compact honogeneous s-contact
manifolds, and for the latter it is possible to extend in a rather straightforward
manner the results established for contact manifolds in [I0 ].
G/K,
(M
Let
mu,
)
(here, and always
be
s-contact manifold of nonzero type.
s
T -principal bundle
is a
p
’
N
Obviously,
G
geneity group
G
since
V
the vertical foliation
_X
3.5, and there
(N, ’)
over the symplectic manifold
with
is also compact and symplectic homogeneous with the homo-
M
preserves the whole structure of
(see Section 3 for notation).
G
ous symplectic manifold with group
6 g
is regular by Proposition
M
Then
N
M
p
in the sequel) a compact homogeneous
X N induced on
PROPOSITION 4. i. The basis
the field
and, particularly,
Now, recall that a homogene-
homogeneous strongly symplectic if for every
is
N
is a Hamiltonian field.
N
of the projection
Then we have
N
M
p
above is homogeneous
strongly symplectic.
[I0 ].
PROOF.
on
N
of
M
that are obviously
(i/al)dm I
mu
(u
X E g
Consider
LXMWl
X
p,XM
N
the basic vertical vector fields
i
Assume, for instance, a # 0, whence
u
G-invariant fields.
(i()) LEuixMo i
ml(XM)
i([E u
,XM])
p*(iXN’)
i
X
i
X
M
i
a
I
i()d
’ -(i/al)dfx
N
COROLLARY 4.2.
i
+
-
i(XM)
i
0
on
N.
d(I(XM))=
and shows that
N
The basis
I
ml
u
fx
projects to a function
we have
which yields
M
on
i,..., s)
Then
1,..., s), and
0
X
M
and the induced fields
(u
and denote, as usual, by E
XN
of the projection
Furthermore, since
p,df
x
is a Hamiltonian field.
p
M
N
(.E.D.
described at the
beginning of this section is a simply connected compact homogeneous symplectlc mani-
fold, and its symplectic form belongs to a Khler metric that is homothetical to a
are zero.
Hodge metric. The Betti numbers
All this is gathered in Theorem I of [|0, p. 341] on the basis of results of
b2h+l(N)
Borel, Lichnerowicz and Milnor.
The above results give us the structure of compact homogeneous s-contact manis
folds of the nonzero type as T -principal bundles over special homogeneous symplectic
manifolds.
Conversely, we have
PROPOSITION 4.3.
Let
p
M
2n+s
N
2n
be a
TS-principal
bundle, where
a compact simply connected homogeneous Hodge manifold with the Khler form
Assume
p
has a connection
(mU)(u
i,..., s)
such that
dmU
aUp,,
N
’
2n
is
where not
532
I. VAISMAN
all
a
u
0
(M,m u
Then
PROOF.
(mu
That
p*’)
)
is a
is a compact homogeneous s-contact manifold
regular s-contact structure is known by Proposition
The homogeneity of this structure will be proven like for the contact case in
I0
I0 ]. Namely, by results of Montgomery and Lichnerowicz as quoted in
we
3.2.
G/K with
N
may assume
G
V X E g
algebra) such that
Poisson
Y
take the Lie algebra
d(0(X)
ks
g
x(M)
IR
g
s
(4. i)
with the zero bracket in ]R
S
where
XN
u=l
X
s
XN
is the horizontal lift of
S
E aU E
u
u= I
(O(X) op)
N
are the constants which appear in Proposition 4.3,
I0,
p. 347] shows that
M.
and define
A
A(x 0)
is transitive on
y[X
Finally, we see again like in
(4 2)
u’
(mu),
(-0(X)
I0
that
e
U
are the basic verticl
u
TxoM,
v
y
im
10
G
that
if we decompose
X(x0)
is such that
horizontal(v)
N, the field
P(X0)
u
u (V))u=
L(X
U
(t u))
follows since the s-structure is of the nonzero type.
the s-structure of
E
and
s
i
G .)
This implies the transitivity of
v
E
we can see like in
u
G
U
Accordingly, there is a connected
Ons
u=l
t
u=l
is a Lie algebra homomorphism and, therefore
y
(M).
Lie group G of left transformations of M, and
acts transitively
M
(l.e., V x 0 E M and
v=horizontal(v) + E U(v) E
and if X
g
which is possible since
E
Then, a computation similar to that
is a finite dimensional Lie subalgebra of
is such that
s
with respect to the connection
vector fields of the s-contact structure of
of
a
by
(t u)
y(X
0
one has
i(XN)’
Now
(N, ’) is
g- P(N) (the
Lie group, and
a compact and semisimple
Hamiltonian space, i.e., there is a Lie algebra homomorphism
0
0
L(X (tu))
and
preserves
This means that
.E..
M
From these results it follows:
COROLLARY 4.4.
Let
(M,) be
a compact homogeneous l.c.s, manifold which admits
2n
T2-principal
2n-2,
N
an invariant t.i.a. Then M has a
bundle structure p
M
where N is a simply connected compact homogeneous Hodge manifold. Conversely, every
such bundle which is endowed with an adequate connection as in Proposition 43 is a
compact homogeneous l.c.s, manifold. If M is as in the present Corollary, its first
Betti number is
bl(M)
Here, only the last
i.
assertion has to be justified, and it follows by first con-
M as a flat circle bundle over a contact manifold P, then
fiberinK P over
as a principal fibre bundle, and finally by applying twice the Gysin exact sequence
theorem and using the fact that
0
sidering
N
bl(N)
Now, let
us consider again a homogeneous l.c.s, manifold (M
hot
be the Lie algebra of G
and g
the subalgebra of those X
Accordingly, we have the exact
say that
field
Xf
M
is
trongly
Hamiltonian l.c.s,
homoseneous
f
g
that
let
g
m()--O.
sequence (3.10).
The symplectic case suggests us to
hor
if for every X
X
is the Hamiltonian
g
M
0 (see Section 2).
hor
if a Lie algebra homomorphism 0
P(M)
g
of a function
G/K, ),
of
P(M)
with
Af
Similarly,
M
is
exists such that
533
LOCALLY CONFORMAL SYMPLECTIC MANIFOLDS
A(im qO)
0
hOT
X E g
and
X
M
one has
a G-invariant t.i.a, then we have the Lie
H -I restricted
and
algebr--a
isomorphism
hot
X E g
defined by
X
M
H
and it has
g.c.s,
is not
_
of Proposition 2.3,
yields a homomorphism showing
is a Hamiltonian l.c.s.
M
that
to fields
M
If
Xq0(X
G/K,) be a compact strongly homogeneous l.c.s, maniPROPOSITION 4.5. Let (M
hoT
is
fold which is not g.c.s., and assume G is connected. Then the Lie algebra g
hot
semlsimple) if A 6
s e span {A} (s
g, and g
semisimple if A
II]
PROOF.
hoT
X 6 g
Every
Proposition 2.1,
hoT
Accordingly, we may define on g
is unique.
f
i..
X,Y
is oriented such that
M
where
,Y
>+ <X,
....
an
]M
a reductive Lie algebra, and that
hot
the centre of g
M
f(X) f (Y) a
(4.3)
GhOr
and
g
Indeed,
a leaf of
ghr-invariant,
Ad
hoT
g
etc.
(4.4)
and
0
m
is semisimple, and
s
Ghr = G
L
0
m
expXlO...exp h’
XI ghor,
G,g
is such that, for instance,
sends the leaf
But then exp XI
is
is
c
whose Lie algebra
of the foliation
g(p), g
q
hot
g
and we conclude that
where
c
s
E g ,then, generally, the situation
XI,...
X2 ghor, X3 hor
f(Y) +
M
acts transitively on every leaf
p,q f L
if
we have
[{f(Z), f (X)}
=-In jM
Now, let us note that there is a connected subgroup
is
an inner product
n
hot
g
Z
and if
0
[Z,YI
That is, the inner product (4.3) is
hoT
and in view of
0
Af
Xf(X_)
satisfies
L
L’,
to a leaf
exp
X_2
X_u, X_v,
X
exp X preserves L" etc., and we also must have some
3
L.
to
leaf)
other
whatever
L"
(or
from
back
us
their
bring
that
exponentials
such
Since any bracket of i.a. is horizontal, if we exchange in g the order of the
exponentials such that X
come next to X X
this adds a factor in G hr
1 2
hr
Then, exp XlOexp
p X =exD X =exp X
preserves L
and is also in G
There--u
-v
fore, eventually, we get some y
such that y(p)
q
This
clearly
.E.D.
implies that V p
L
and
T L there is an element X E g hot such that
P
P
sends
L’
to
L"
,X_v,X_w
X2=ex
Ghor-W
p
XM(P}
Furthermore, let us look at the previous decomposition ghoT
sider X
c i.e., [X,Y]
0 for all Y ghot
This implies
{f (X), f(Y)}
hence
ment shows that
i(XM)
i.e.,
---(,YM
(i(XM))p (p)
0
Vp
%
M
VSp TpM
and
A.
and
and con-
c
X{f(X),
This equality together with the
0
%m for some function
s
0
f(Y)}
p--reviou--s argu-
such that
..D.
mp(p)=0,
Let us note the following Corollary which, as a matter of fact, follows also from
the proof of Proposition 4.3.
COROLLARY 4.6.
Let
(M=G/K,)
manifold with an invariant t.i.a.B.
hoT
g
/
can be also represented as
g
s @
2
g
g
span {A}span {B}
By Proposition 4.5
but not
M
is of the form
span {A}
follows.
Then
Consider the Lie algebra
the Lie algebra of
PROOF.
be a compact homogeneous l.c.s,
hor
we have g
and
s__
s__
/
g.c.s.
where
is a semisimple Lie algebra.
@ span
{A},
Clearly,
hor
g
and the result
I. VAISMAN
534
Another related result is
G/K,) be a compact strongly homogeneous l.c.s, but
g.c.s, manifold such that A doesn’t represent an element of g. Then M has
COROLLARY 4.7.
not
(M
Let
transversal infinitesimal automorphism B
PROOF. We know that the Lie algebra g is not semisimple, and, therefore, it
hor
is a commutative ideal of
g
must have a nonzero abelian ideal I. Then, I
hor
hor
hor
O. Hence
g
is semisimple, I
and since, under the hypothesis,
a G-invariant
dim I
i, and
[X,B]
is an ideal we have
I
[X,B]
0
and
%B, V X 6 g
and since
(B)
and
Since
i.
[X,B]
has to be horizontal,
B
is the requested i.a.
But then
g
is a central element in
B
B
B where
I can be seen as generated by
Q...
RIEMANNIAN MANIFOLDS
In the present paper, we used the Boothby-Wang fibration technique of
5.
8
in order to clarify the geometric structure of a regular l.c.s, manifold.
This
is an interesting technique, and we should like to indicate here a different application of it.
Let
This
section is not on l.c.s, manifolds but on Riemannian manifolds.
m
M
be a compact connected Riemannian manifold with the metric g
Let us
s
m
assume that there is given an action of the additive group IR
on M
by isometries
of
g
all of whose orbits are s-dimensional.
on
M
a foliation
V
Then, the orbits of this action define
(called the vertical foliation) whose leaves are s-dimensional
submanifolds tangent to some independent commuting vector fields
provided by the natural basis of
]R
s
Clearly, we have
u
g
E
u
(u
0
i,..., s)
If
V
is a
simple foliation whose space of leaves is Hausdorff, we say that the action of
M
s
IR
on
is regular.
A few more simple details about
(M,g) and the action above will be needed.
C be the horizontal distribution orthogonal to
Then, we can define
Namely, let
the 2-tensor
y(X,Y)
g(pr C X, pr C Y)
and the s 1-forms
u
u
(E)
V
Since every vector field
X
/C
0
(u,v =1,
s)
(5.2)
has a unique decomposition
s
X
u
V
mU(x)
X’ + Z
E
u= I
X’ E C
U
we get
S
g(X,Y)
Furthermore since
pr
C
E
u
y(X,Y) +
preserve
Z
u,v=l
g
g(Eu,Ev)mU(x)mV(Y)
and
(5.4)
they also preserve
C
and commute with
Hence
eE
Conversely, let
(Mm,y,mu)
y
u
(u=l
v
0
O.
(5.5)
u
s) be a differentiable manifold endowed with
a positive semidefinite 2-covariant tensor y of rank m-s
u
Pfaff forms
Then
0 defines a subbundle C of
mu
defines a subbundle
Assume that the structure is
and with
s
independent
TM, and {X/i(X)y
such that TM
C
O}
and
535
LOCALLY CONFORMAL SYMPLECTIC MANIFOLDS
E
define vector fields
in
u
V such
the following relations hold
E=o,
u
L
vu
V(E u)
that
E--
L
assume that
Furthermore,
(5.6)
O.
u
and use (4) in order to get a Riemannian metric
Then, we may define
we shall have
admitting E u as Killing vector fields. Furthermore,
6uv
g(Eu,Ev)
(LEu mV)(Ew)
(L
0
e uy)(Ev,X’
0
_V([Eu ,Ev])
=-([Eu,Evl, X’)
(X’ E ),
Hence, if the structure (,, u) satis1,..., s
/ and (6) it provides M with a Riemannian structure, and an isetric
fies
s
with s-dimensional orbits.
action of 1t
theore
Now, we can formulate the folIowing Boothby-Wang type fibration
Let (Mm,g) be a compact connected Riemannian manifold endowed
whence
[E ,E
C
’IN
0
u,v
for all
PROPOSITION.
with a regular isometric action of
s
with the associated structure
Then, the projection
above, and with the vertical foliation
a TS(torus)-principal bundle endowed with a connection
basis
B
y’
has a Riemannian metric
submersion.
Conversely, if
p
M
such that
B
is a
p
(CuOu)(C u
y, and
TS-principal
p
p
(y,mu)
M
B
const.).
defined
M/
is
The
is a Riemannian
bundle over the Riemannian
(mu)
is a connection of this bundle then M admits a R[emanmanifold (B,y’), and
s
nian metric and a regular isometric action of IR with s-dimensional orbits such that
is a Riemannian submersion.
p
The proof of the existence of the principal bundle structure required is
exactly the same as in the case of Proposition 3.1. All the other facts stated in
PROOF.
Proposition are easy consequences of the formulas (i)
(6).
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LIBERMANN, P.
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