Calculus on symplectic manifolds. (English) Zbl 1463.53085

Eastwood, Michael; Slovák, Jan

Arch. Math., Brno 54, No. 5, 265-280 (2018).

The main result of the article is the following.
Theorem. If \(M^{2n}\) is a symplectic manifold and \(\nabla_a\) is its symplectic connection, then there is a natural vector bundle on \(M^{2n}\) of rank \(2(n+1)\) equipped with a connection, which is symplectically flat if and only if the Riemannian curvature tensor \(R_{ab}{}^c{}_d\) of \(\nabla_a\) has a special form:\[R_{ab}{}^c{}_d=\delta_a^c\,P_{bd}-\delta_b^c\,P_{ad}+J_{ad}\,P_{be}\,J^{ce}-J_{bd}\,P_{ae}\,J^{ce}+2J_{ab}\,P_{de}\,J^{ce}\] where \(P_{ab}\) is a symmetric tensor.
Some important particular cases when \(P_{ab}\) is the standard metric and more generally, when the symplectic connection arises from a Kählerian metric are studied. It is proved that a symplectic tractor connection on a Kählerian manifold is symplectically flat if and only if the metric of the manifold has constant holomorphic sectional curvature.
On complex projective space \(\mathbb{C}P_n\), a Bernstein-Gelfand-Gelfand-like series of differential complexes is constructed.

Reviewer: Mihail Banaru (Smolensk)

MSC:

53D05	Symplectic manifolds (general theory)
53B35	Local differential geometry of Hermitian and Kählerian structures

Keywords:

symplectic structure; Kählerian structure; complex projective space; BGG-complex; symplectic connection; tractor calculus

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