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**Topological invariants of connections on symplectic manifolds.**
*(English.
Russian original)*
Zbl 0878.58002

Funct. Anal. Appl. 29, No. 4, 258-267 (1995); translation from Funkts. Anal. Prilozh. 29, No. 4, 45-56 (1995).

From the introduction: “Let \(M^{2n}\) be a \(2n\)-dimensional symplectic manifold, let \(E\to M\) be a vector bundle whose structure group is a connected semisimple Lie group \(G\), let \(\nabla^M\) be a symplectic connection on \(M\), and let \(\nabla^E\) be a connection on \(E\).

This paper is devoted to the problem of finding all closed differential forms on \(M\) that can be written in local Darboux coordinates as polynomials in finite-order derivatives of the coefficients of the connections \(\nabla^E\) and \(\nabla^M\) under the condition that the cohomology class of the manifold \(M\) defined by such a form is preserved under the deformations of the connections. It is required that this differential form be well defined on \(M\). This is possible only for the case in which the dependence of this form on the connection coefficients is preserved under the transformations of Darboux coordinates. These forms are said to be invariant.

A similar problem for Riemannian manifolds was solved by A. A. Abramov [Dokl. Akad. Nauk SSSR, n. Ser. 81, 125-128 (1951; Zbl 0044.36901)] [see also P. Gilkey, ‘Invariance theory, the heat equation, and the Atiyah-Singer index theorem’ (1984; Zbl 0565.58035)]. For a more detailed statement of this problem and its solution see M. Atiyah, R. Bott, and V. K. Patody [Invent. Math. 19, 279-330 (1973; Zbl 0257.58008)].

By analogy with the case of Riemannian manifolds, every invariant form on \(M\) is a polynomial in Pontryagin classes of the manifold \(M\), characteristic classes of the bundle \(E\), and the symplectic form \(\omega\) (up to the so-called trivial forms whose cohomology classes are always trivial)”.

This paper is devoted to the problem of finding all closed differential forms on \(M\) that can be written in local Darboux coordinates as polynomials in finite-order derivatives of the coefficients of the connections \(\nabla^E\) and \(\nabla^M\) under the condition that the cohomology class of the manifold \(M\) defined by such a form is preserved under the deformations of the connections. It is required that this differential form be well defined on \(M\). This is possible only for the case in which the dependence of this form on the connection coefficients is preserved under the transformations of Darboux coordinates. These forms are said to be invariant.

A similar problem for Riemannian manifolds was solved by A. A. Abramov [Dokl. Akad. Nauk SSSR, n. Ser. 81, 125-128 (1951; Zbl 0044.36901)] [see also P. Gilkey, ‘Invariance theory, the heat equation, and the Atiyah-Singer index theorem’ (1984; Zbl 0565.58035)]. For a more detailed statement of this problem and its solution see M. Atiyah, R. Bott, and V. K. Patody [Invent. Math. 19, 279-330 (1973; Zbl 0257.58008)].

By analogy with the case of Riemannian manifolds, every invariant form on \(M\) is a polynomial in Pontryagin classes of the manifold \(M\), characteristic classes of the bundle \(E\), and the symplectic form \(\omega\) (up to the so-called trivial forms whose cohomology classes are always trivial)”.

Reviewer: A.Iozzi (College Park)

### MSC:

58A10 | Differential forms in global analysis |

53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |

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\textit{D. E. Tamarkin}, Funct. Anal. Appl. 29, No. 4, 258--267 (1995; Zbl 0878.58002); translation from Funkts. Anal. Prilozh. 29, No. 4, 45--56 (1995)

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### References:

[1] | A. A. Abramov, ?Topological invariants of Riemannian spaces obtained by integration of tensor fields,? Dokl. Akad. Nauk SSSR,81 (1951). |

[2] | P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah?Singer Index Theorem, Math. Lect. Series, Vol. 11, Publish of Perish Inc., Wilmington, Del. (1984). · Zbl 0565.58035 |

[3] | M. F. Atiyah, R. Bott, and V. K. Patody, ?On the heat equation and the index theorem,? Invent. Math.,19, 279-330 (1973). · Zbl 0257.58008 |

[4] | D. B. Fuks, Cohomology of Infinite-Dimensional Lie Algebras [in Russian], Nauka, Moscow (1984). · Zbl 0592.17011 |

[5] | P. Olver, Application of Lie Groups to Differential Equations, Springer-Verlag, New York?Berlin?Heidelberg?Tokyo (1986). · Zbl 0588.22001 |

[6] | R. M. Switzer, Algebraic Topology ? Homotopy and Homology, Springer-Verlag, Berlin a.o. (1975). · Zbl 0305.55001 |

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