Almost-symplectic structure

A non-degenerate differential -form on a manifold. An almost-symplectic structure can exist only on an even-dimensional manifold () and defines an -structure , namely the principal fibre bundle of frames on with structure group , consisting of all frames for which

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

An almost-complex structure and a Riemannian metric on a manifold define an almost-symplectic structure by the formula

where and 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 in some local coordinates , in a neighbourhood of any point. According to Darboux's theorem, for this it is necessary and sufficient that be closed. An example of an integrable almost-symplectic structure is the canonical symplectic structure on the cotangent bundle of an arbitrary manifold (here the are local coordinates on and the are the associated coordinates in the fibres). An example of a non-integrable almost-symplectic structure is a left-invariant -form on a semi-simple Lie group , obtained by extending an arbitrary non-degenerate exterior -form on the Lie algebra of by left translation to . 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 -form , called its volume form, and several operators in the space of differential forms: the operator of exterior multiplication by ; the operator of interior multiplication by ; the Hodge star operator , , where the operator of interior multiplication is defined as the contraction of the given form with the -vector corresponding to the -form ; the operator of codifferentiation . In contrast with the Riemannian case, the operator turns out to be skew-symmetric with respect to the global scalar product in the space of -forms on a compact manifold . For an arbitrary -form one has the Hodge–Lepage decomposition , where the are uniquely determined effective forms (i.e. they are annihilated by ) [3].

An almost-symplectic structure is said to be conformally flat if there is a function such that . This is equivalent to the representability of in the form

For , a necessary and sufficient condition in order that the almost-symplectic structure be conformally flat is the closedness of the -form , and for the equality should hold (cf. [1]).

The tensor of type corresponding to the -form and defined by the equality , where and are vectors, is called the torsion tensor of the almost-symplectic structure . The (degenerate) metric can be associated with it. An almost-symplectic structure determines the class of linear connections for which is parallel and which have as their torsion tensor. Two such connections differ by a tensor field of the form , where is an arbitrary symmetric tensor field. The connections under consideration correspond in a one-to-one manner to the sections of the first extension for the -structure , which is the principal bundle of frames on with structure group (the vector group of homogeneous polynomials in variables of degree 3). The -structure is a -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 -transitive group for any .

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)