View source for Almost-symplectic structure

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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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]]]).

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

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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-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

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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]]]).

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" />.

====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>



====Comments====


====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>