# 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