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A non-degenerate differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a0120101.png" />-form on a manifold. An almost-symplectic structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a0120102.png" /> can exist only on an even-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a0120103.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a0120104.png" />) and defines an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a0120105.png" />-structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a0120106.png" />, namely the principal fibre bundle of frames on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a0120107.png" /> with structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a0120108.png" />, consisting of all frames <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a0120109.png" /> for which
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201010.png" /></td> </tr></table>
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A necessary and sufficient condition for the existence of an almost-symplectic structure (or of an almost-complex structure, as well) on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201011.png" /> is the possibility of reducing the structure group of the tangent bundle to the unitary group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201012.png" />. For this, in particular, it is necessary that all odd-dimensional Stiefel–Whitney classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201013.png" /> vanish (cf. [[#References|[1]]]).
+
A non-degenerate differential  $  2 $-
 +
form on a manifold. An almost-symplectic structure $  \Omega $
 +
can exist only on an even-dimensional manifold  $  M $(
 +
$  \mathop{\rm dim}  M = 2 m $)
 +
and defines an $  \mathop{\rm Sp} ( m , \mathbf R ) $-
 +
structure $  B _ { \mathop{\rm Sp}  ( m , \mathbf R ) } $,
 +
namely the principal fibre bundle of frames on  $  M $
 +
with structure group $  \mathop{\rm Sp} ( m , \mathbf R ) $,  
 +
consisting of all frames  $  r = \{ {e _ {i} , f _ {i} } : {i = 1 \dots m } \} $
 +
for which
  
An almost-complex structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201014.png" /> and a Riemannian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201015.png" /> on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201016.png" /> define an almost-symplectic structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201017.png" /> by the formula
+
$$
 +
\Omega ( e _ {i} , e _ {j} )  = \
 +
\Omega ( f _ {i} , f _ {j} )  = \
 +
0 ,\  \Omega
 +
( e _ {i} , f _ {j} )  = \
 +
\delta _ {ij} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201018.png" /></td> </tr></table>
+
A necessary and sufficient condition for the existence of an almost-symplectic structure (or of an almost-complex structure, as well) on a manifold  $  M $
 +
is the possibility of reducing the structure group of the tangent bundle to the unitary group  $  U (m) $.  
 +
For this, in particular, it is necessary that all odd-dimensional Stiefel–Whitney classes of  $  M $
 +
vanish (cf. [[#References|[1]]]).
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201020.png" /> are vectors. Any almost-symplectic structure can be obtained in this manner. An almost-symplectic structure is said to be integrable or, in other words, a symplectic structure, if it can be brought to the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201021.png" /> in some local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201022.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201023.png" />, in a neighbourhood of any point. According to Darboux's theorem, for this it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201024.png" /> be closed. An example of an integrable almost-symplectic structure is the canonical symplectic structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201025.png" /> on the cotangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201026.png" /> of an arbitrary manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201027.png" /> (here the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201028.png" /> are local coordinates on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201029.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201030.png" /> are the associated coordinates in the fibres). An example of a non-integrable almost-symplectic structure is a left-invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201031.png" />-form on a semi-simple Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201032.png" />, obtained by extending an arbitrary non-degenerate exterior <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201033.png" />-form on the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201034.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201035.png" /> by left translation to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201036.png" />. As a Riemannian metric, an almost-symplectic structure also defines an isomorphism of the tangent and cotangent spaces (and by the same method, of the spaces of contravariant and covariant tensors); it further defines a canonical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201037.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201038.png" />, called its volume form, and several operators in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201039.png" /> of differential forms: the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201040.png" /> of exterior multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201041.png" />; the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201042.png" /> of interior multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201043.png" />; the Hodge star operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201045.png" />, where the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201046.png" /> of interior multiplication is defined as the contraction of the given form with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201047.png" />-vector corresponding to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201048.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201049.png" />; the operator of codifferentiation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201050.png" />. In contrast with the Riemannian case, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201051.png" /> turns out to be skew-symmetric with respect to the global scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201052.png" /> in the space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201053.png" />-forms on a compact manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201054.png" />. For an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201055.png" />-form one has the Hodge–Lepage decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201056.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201057.png" /> are uniquely determined effective forms (i.e. they are annihilated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201058.png" />) [[#References|[3]]].
+
An almost-complex structure $  J $
 +
and a Riemannian metric  $  g $
 +
on a manifold $  M $
 +
define an almost-symplectic structure $  \Omega $
 +
by the formula
  
An almost-symplectic structure is said to be conformally flat if there is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201059.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201060.png" />. This is equivalent to the representability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201061.png" /> in the form
+
$$
 +
\Omega ( X , Y )  =  g
 +
( J X , Y ) - g
 +
( X , J Y ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201062.png" /></td> </tr></table>
+
where  $  X $
 +
and  $  Y $
 +
are vectors. Any almost-symplectic structure can be obtained in this manner. An almost-symplectic structure is said to be integrable or, in other words, a symplectic structure, if it can be brought to the form  $  \Omega = \sum d x  ^ {i} \wedge d y  ^ {i} $
 +
in some local coordinates  $  x  ^ {i} , y  ^ {i} , $
 +
$  i = 1 \dots m $,
 +
in a neighbourhood of any point. According to Darboux's theorem, for this it is necessary and sufficient that  $  \Omega $
 +
be closed. An example of an integrable almost-symplectic structure is the canonical symplectic structure  $  \Omega = \sum d p  ^ {i} \wedge d q  ^ {i} $
 +
on the cotangent bundle  $  T  ^ {*} M $
 +
of an arbitrary manifold  $  M $(
 +
here the  $  q  ^ {i} $
 +
are local coordinates on  $  M $
 +
and the  $  p  ^ {i} $
 +
are the associated coordinates in the fibres). An example of a non-integrable almost-symplectic structure is a left-invariant  $  2 $-
 +
form on a semi-simple Lie group  $  G $,
 +
obtained by extending an arbitrary non-degenerate exterior  $  2 $-
 +
form on the Lie algebra  $  T _ {e} G $
 +
of  $  G $
 +
by left translation to  $  G $.
 +
As a Riemannian metric, an almost-symplectic structure also defines an isomorphism of the tangent and cotangent spaces (and by the same method, of the spaces of contravariant and covariant tensors); it further defines a canonical  $  2m $-
 +
form  $  \eta = \Omega  ^ {m} / m ! $,
 +
called its volume form, and several operators in the space  $  \wedge (M) $
 +
of differential forms: the operator  $  \epsilon _  \Omega  $
 +
of exterior multiplication by  $  \Omega $;
 +
the operator  $  i _  \Omega  $
 +
of interior multiplication by  $  \Omega $;  
 +
the Hodge star operator  $  * :  \wedge  ^ {p} (M) \rightarrow \wedge  ^ {2m-p} (M) $,
 +
$  \omega \rightarrow i _  \omega  \eta $,
 +
where the operator  $  i _  \omega  $
 +
of interior multiplication is defined as the contraction of the given form with the  $  p $-
 +
vector corresponding to the  $  p $-
 +
form  $  \omega $;  
 +
the operator of codifferentiation  $  \delta = * d * $.
 +
In contrast with the Riemannian case, the operator  $  \Delta = d \delta + \delta d $
 +
turns out to be skew-symmetric with respect to the global scalar product  $  \langle  \alpha , \beta \rangle \int _ {M} \alpha \wedge * \beta $
 +
in the space of  $  p $-
 +
forms on a compact manifold  $  M $.
 +
For an arbitrary  $  p $-
 +
form one has the Hodge–Lepage decomposition  $  \omega = \omega _ {0} + \epsilon _  \Omega  \omega _ {1} + \epsilon _  \Omega  ^ {2} \omega _ {2} + \dots $,
 +
where the  $  \omega _ {i} \in \Lambda  ^ {p-2i} (M) $
 +
are uniquely determined effective forms (i.e. they are annihilated by  $  i _  \Omega  $)
 +
[[#References|[3]]].
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201063.png" />, a necessary and sufficient condition in order that the almost-symplectic structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201064.png" /> be conformally flat is the closedness of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201065.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201066.png" />, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201067.png" /> the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201068.png" /> should hold (cf. [[#References|[1]]]).
+
An almost-symplectic structure is said to be conformally flat if there is a function  $  \lambda > 0 $
 +
such that  $  d ( \lambda \Omega ) = 0 $.  
 +
This is equivalent to the representability of  $  \Omega $
 +
in the form
  
The tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201069.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201070.png" /> corresponding to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201071.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201072.png" /> and defined by the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201073.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201075.png" /> are vectors, is called the torsion tensor of the almost-symplectic structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201076.png" />. The (degenerate) metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201077.png" /> can be associated with it. An almost-symplectic structure determines the class of linear connections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201078.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201079.png" /> is parallel and which have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201080.png" /> as their torsion tensor. Two such connections differ by a tensor field of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201081.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201082.png" /> is an arbitrary symmetric tensor field. The connections under consideration correspond in a one-to-one manner to the sections of the first extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201083.png" /> for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201084.png" />-structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201085.png" />, which is the principal bundle of frames on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201086.png" /> with structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201087.png" /> (the vector group of homogeneous polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201088.png" /> variables of degree 3). The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201089.png" />-structure is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201090.png" />-structure of infinite type. Therefore, the group of automorphisms of an almost-symplectic structure can be infinite-dimensional. In particular, the group of automorphisms of a symplectic structure is always infinite-dimensional and is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201091.png" />-transitive group for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012010/a01201092.png" />.
+
$$
 +
\Omega  = y  ^ {1}
 +
\sum _ { i=1 } ^ { m }
 +
d x  ^ {i} \wedge
 +
d y  ^ {i} .
 +
$$
 +
 
 +
For  $  m = 2 $,
 +
a necessary and sufficient condition in order that the almost-symplectic structure  $  \Omega $
 +
be conformally flat is the closedness of the  $  1 $-
 +
form  $  \delta \Omega = i _  \Omega  d \Omega $,
 +
and for  $  m > 2 $
 +
the equality  $  d \Omega = ( 1 / m - 1 ) \delta \Omega \wedge \Omega $
 +
should hold (cf. [[#References|[1]]]).
 +
 
 +
The tensor  $  T $
 +
of type  $  ( 1 , 2 ) $
 +
corresponding to the $  3 $-
 +
form $  d \Omega $
 +
and defined by the equality $  \Omega ( T _ {X} Y , Z ) = d \Omega ( X , Y , Z ) $,  
 +
where $  X , Y $
 +
and $  Z $
 +
are vectors, is called the torsion tensor of the almost-symplectic structure $  \Omega $.  
 +
The (degenerate) metric $  g ( X , Y ) = \mathop{\rm tr}  T _ {X} T _ {Y} $
 +
can be associated with it. An almost-symplectic structure determines the class of linear connections $  \nabla $
 +
for which $  \Omega $
 +
is parallel and which have $  T $
 +
as their torsion tensor. Two such connections differ by a tensor field of the form $  \Omega  ^ {ij} S _ {jkl} $,  
 +
where $  S _ {jkl} $
 +
is an arbitrary symmetric tensor field. The connections under consideration correspond in a one-to-one manner to the sections of the first extension $  B  ^ {1} \rightarrow B $
 +
for the $  \mathop{\rm Sp} ( m , \mathbf R ) $-
 +
structure $  B = B _ { \mathop{\rm Sp}  ( m , \mathbf R ) } $,  
 +
which is the principal bundle of frames on $  B $
 +
with structure group $  S  ^ {3} ( \mathbf R  ^ {2m} ) $(
 +
the vector group of homogeneous polynomials in $  2 m $
 +
variables of degree 3). The $  \mathop{\rm Sp} ( m , \mathbf R ) $-
 +
structure is a $  G $-
 +
structure of infinite type. Therefore, the group of automorphisms of an almost-symplectic structure can be infinite-dimensional. In particular, the group of automorphisms of a symplectic structure is always infinite-dimensional and is a $  k $-
 +
transitive group for any $  k > 0 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Liberman,  "Sur les structures presque complexe et autres structures infinitésimales régulières"  ''Bull. Soc. Math. France'' , '''83'''  (1955)  pp. 195–224</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  ''Itogi Nauk i Tekhn. Algebra Topol. Geom.'' , '''11'''  (1974)  pp. 153–207</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.V. Lychagin,  "Contact geometry and second-order non-linear differential equations"  ''Russian Math. Surveys'' , '''34''' :  1  (1979)  pp. 149–180  ''Uspekhi Mat. Nauk'' , '''34''' :  1  (1979)  pp. 137–165</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Kobayashi,  "Transformation groups in differential geometry" , Springer  (1972)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.E. Hurt,  "Geometric quantization in action" , Reidel  (1983)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.I. Arnol'd,  A.B. Givental,  ''Itogi Nauk. i Tekhn. Sovrem. Probl. Mat.'' , '''4'''  pp. 5–139</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Liberman,  "Sur les structures presque complexe et autres structures infinitésimales régulières"  ''Bull. Soc. Math. France'' , '''83'''  (1955)  pp. 195–224</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  ''Itogi Nauk i Tekhn. Algebra Topol. Geom.'' , '''11'''  (1974)  pp. 153–207</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.V. Lychagin,  "Contact geometry and second-order non-linear differential equations"  ''Russian Math. Surveys'' , '''34''' :  1  (1979)  pp. 149–180  ''Uspekhi Mat. Nauk'' , '''34''' :  1  (1979)  pp. 137–165</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Kobayashi,  "Transformation groups in differential geometry" , Springer  (1972)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.E. Hurt,  "Geometric quantization in action" , Reidel  (1983)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.I. Arnol'd,  A.B. Givental,  ''Itogi Nauk. i Tekhn. Sovrem. Probl. Mat.'' , '''4'''  pp. 5–139</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Libermann,  C.-M. Marle,  "Symplectic geometry and analytical mechanics" , Reidel  (1987)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Libermann,  C.-M. Marle,  "Symplectic geometry and analytical mechanics" , Reidel  (1987)  (Translated from French)</TD></TR></table>

Latest revision as of 16:10, 1 April 2020


A non-degenerate differential $ 2 $- form on a manifold. An almost-symplectic structure $ \Omega $ can exist only on an even-dimensional manifold $ M $( $ \mathop{\rm dim} M = 2 m $) and defines an $ \mathop{\rm Sp} ( m , \mathbf R ) $- structure $ B _ { \mathop{\rm Sp} ( m , \mathbf R ) } $, namely the principal fibre bundle of frames on $ M $ with structure group $ \mathop{\rm Sp} ( m , \mathbf R ) $, consisting of all frames $ r = \{ {e _ {i} , f _ {i} } : {i = 1 \dots m } \} $ for which

$$ \Omega ( e _ {i} , e _ {j} ) = \ \Omega ( f _ {i} , f _ {j} ) = \ 0 ,\ \Omega ( e _ {i} , f _ {j} ) = \ \delta _ {ij} , $$

A necessary and sufficient condition for the existence of an almost-symplectic structure (or of an almost-complex structure, as well) on a manifold $ M $ is the possibility of reducing the structure group of the tangent bundle to the unitary group $ U (m) $. For this, in particular, it is necessary that all odd-dimensional Stiefel–Whitney classes of $ M $ vanish (cf. [1]).

An almost-complex structure $ J $ and a Riemannian metric $ g $ on a manifold $ M $ define an almost-symplectic structure $ \Omega $ by the formula

$$ \Omega ( X , Y ) = g ( J X , Y ) - g ( X , J Y ) , $$

where $ X $ and $ Y $ are vectors. Any almost-symplectic structure can be obtained in this manner. An almost-symplectic structure is said to be integrable or, in other words, a symplectic structure, if it can be brought to the form $ \Omega = \sum d x ^ {i} \wedge d y ^ {i} $ in some local coordinates $ x ^ {i} , y ^ {i} , $ $ i = 1 \dots m $, in a neighbourhood of any point. According to Darboux's theorem, for this it is necessary and sufficient that $ \Omega $ be closed. An example of an integrable almost-symplectic structure is the canonical symplectic structure $ \Omega = \sum d p ^ {i} \wedge d q ^ {i} $ on the cotangent bundle $ T ^ {*} M $ of an arbitrary manifold $ M $( here the $ q ^ {i} $ are local coordinates on $ M $ and the $ p ^ {i} $ are the associated coordinates in the fibres). An example of a non-integrable almost-symplectic structure is a left-invariant $ 2 $- form on a semi-simple Lie group $ G $, obtained by extending an arbitrary non-degenerate exterior $ 2 $- form on the Lie algebra $ T _ {e} G $ of $ G $ by left translation to $ G $. As a Riemannian metric, an almost-symplectic structure also defines an isomorphism of the tangent and cotangent spaces (and by the same method, of the spaces of contravariant and covariant tensors); it further defines a canonical $ 2m $- form $ \eta = \Omega ^ {m} / m ! $, called its volume form, and several operators in the space $ \wedge (M) $ of differential forms: the operator $ \epsilon _ \Omega $ of exterior multiplication by $ \Omega $; the operator $ i _ \Omega $ of interior multiplication by $ \Omega $; the Hodge star operator $ * : \wedge ^ {p} (M) \rightarrow \wedge ^ {2m-p} (M) $, $ \omega \rightarrow i _ \omega \eta $, where the operator $ i _ \omega $ of interior multiplication is defined as the contraction of the given form with the $ p $- vector corresponding to the $ p $- form $ \omega $; the operator of codifferentiation $ \delta = * d * $. In contrast with the Riemannian case, the operator $ \Delta = d \delta + \delta d $ turns out to be skew-symmetric with respect to the global scalar product $ \langle \alpha , \beta \rangle \int _ {M} \alpha \wedge * \beta $ in the space of $ p $- forms on a compact manifold $ M $. For an arbitrary $ p $- form one has the Hodge–Lepage decomposition $ \omega = \omega _ {0} + \epsilon _ \Omega \omega _ {1} + \epsilon _ \Omega ^ {2} \omega _ {2} + \dots $, where the $ \omega _ {i} \in \Lambda ^ {p-2i} (M) $ are uniquely determined effective forms (i.e. they are annihilated by $ i _ \Omega $) [3].

An almost-symplectic structure is said to be conformally flat if there is a function $ \lambda > 0 $ such that $ d ( \lambda \Omega ) = 0 $. This is equivalent to the representability of $ \Omega $ in the form

$$ \Omega = y ^ {1} \sum _ { i=1 } ^ { m } d x ^ {i} \wedge d y ^ {i} . $$

For $ m = 2 $, a necessary and sufficient condition in order that the almost-symplectic structure $ \Omega $ be conformally flat is the closedness of the $ 1 $- form $ \delta \Omega = i _ \Omega d \Omega $, and for $ m > 2 $ the equality $ d \Omega = ( 1 / m - 1 ) \delta \Omega \wedge \Omega $ should hold (cf. [1]).

The tensor $ T $ of type $ ( 1 , 2 ) $ corresponding to the $ 3 $- form $ d \Omega $ and defined by the equality $ \Omega ( T _ {X} Y , Z ) = d \Omega ( X , Y , Z ) $, where $ X , Y $ and $ Z $ are vectors, is called the torsion tensor of the almost-symplectic structure $ \Omega $. The (degenerate) metric $ g ( X , Y ) = \mathop{\rm tr} T _ {X} T _ {Y} $ can be associated with it. An almost-symplectic structure determines the class of linear connections $ \nabla $ for which $ \Omega $ is parallel and which have $ T $ as their torsion tensor. Two such connections differ by a tensor field of the form $ \Omega ^ {ij} S _ {jkl} $, where $ S _ {jkl} $ is an arbitrary symmetric tensor field. The connections under consideration correspond in a one-to-one manner to the sections of the first extension $ B ^ {1} \rightarrow B $ for the $ \mathop{\rm Sp} ( m , \mathbf R ) $- structure $ B = B _ { \mathop{\rm Sp} ( m , \mathbf R ) } $, which is the principal bundle of frames on $ B $ with structure group $ S ^ {3} ( \mathbf R ^ {2m} ) $( the vector group of homogeneous polynomials in $ 2 m $ variables of degree 3). The $ \mathop{\rm Sp} ( m , \mathbf R ) $- structure is a $ G $- structure of infinite type. Therefore, the group of automorphisms of an almost-symplectic structure can be infinite-dimensional. In particular, the group of automorphisms of a symplectic structure is always infinite-dimensional and is a $ k $- transitive group for any $ k > 0 $.

References

[1] P. Liberman, "Sur les structures presque complexe et autres structures infinitésimales régulières" Bull. Soc. Math. France , 83 (1955) pp. 195–224
[2] Itogi Nauk i Tekhn. Algebra Topol. Geom. , 11 (1974) pp. 153–207
[3] V.V. Lychagin, "Contact geometry and second-order non-linear differential equations" Russian Math. Surveys , 34 : 1 (1979) pp. 149–180 Uspekhi Mat. Nauk , 34 : 1 (1979) pp. 137–165
[4] S. Kobayashi, "Transformation groups in differential geometry" , Springer (1972)
[5] N.E. Hurt, "Geometric quantization in action" , Reidel (1983)
[6] V.I. Arnol'd, A.B. Givental, Itogi Nauk. i Tekhn. Sovrem. Probl. Mat. , 4 pp. 5–139

Comments

References

[a1] P. Libermann, C.-M. Marle, "Symplectic geometry and analytical mechanics" , Reidel (1987) (Translated from French)
How to Cite This Entry:
Almost-symplectic structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Almost-symplectic_structure&oldid=45087
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article