Almost-symplectic structure

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

References

[a1]	P. Libermann, C.-M. Marle, "Symplectic geometry and analytical mechanics" , Reidel (1987) (Translated from French)