Abstract
In the paper, one establishes the decomposition of the space of tensors which have the symmetries of the curvature of a torsionless symplectic connection into Sp (n)-irreducible components. This leads to three interesting classes of symplectic connections: flat, Ricci flat, and similar to the Levi-Civita connections of Kähler manifolds with constant holomorphic sectional curvature (we call them connections with reducible curvature). A symplectic manifold with two transversal polarizations has a canonical symplectic connection, and we study properties that are encountered if this canonical connection belongs to the classes mentioned above. For instance, in the reducible case we can compute the Pontrjagin classes, and these will be zero if the polarizations are real, etc. If the polarizations are real and there exist points where they are either singular or nontransversal, one has residues in the sense ofLehmann [L], which should be expected to play an interesting role in symplectic geometry.
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References
[BGM]Berger, M., Gauduchon, P., Mazet, E.: Le spectre d'une variété riemannienne. Lect. Notes Math.194. Berlin-Heidelberg-New York: Springer. 1971.
[D]Deheuvels, R.: Formes Quadratiques et Groupes Classiques. Paris: Presses Univ. de France. 1981.
[H]Hess, H.: Connections on symplectic manifolds and geometric quantization. In: Differential Geometric Methods in Math. Physics. Proc. Conf. Aix-en-Province 1979. Lect. Notes Math.836, 153–166. Berlin-Heidelberg-New York: Springer.
[KN]Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry I, II. New York: Intersc. Publ. 1963, 1969.
[L]Lehmann, D.: Résidus des connexions à singularités et classes carácteristiques. Ann. Inst. Fourier,31, 83–98 (1981).
[Lb]Libermann, P.: Sur le problème d'équivalence de certaines structures infinitésimales. Ann. Mat. Pura Appl.36, 27–120 (1954).
[Lb′]Libermann, P.: Problèmes d'équivalence et géométrie symplectique. Astérisque107–108, 43–68 (1983).
[Li]Lichnerowicz, A.: Quantum Mechanics and Deformations of Geometrical Dynamics. In: Quantum Theory, Groups, Fields and Particles. (Barut, A. O., ed.), pp. 3–82. Dordrecht: Reidel, 1983.
[M]Morvan, J. M.: Quelques invariants topologiques en géométrie symplectique. Ann. Inst. H. Poincaré,38, 349–370 (1983).
[N]Nirenberg, L.: A Complex Frobenius theorem. Sem. Anal. Funct.1, 172–189 (1958).
[R]Reinhart, B.: Harmonic integrals on foliated manifolds. American J. Math.81, 529–539 (1959).
[ST]Singer, I. M., Thorpe, J. A.: The curvature of 4-dimensional Einstein spaces. Global Analysis (papers in honor of K. Kodaira), pp. 355–365. Tokyo: Univ. of Tokyo Press. 1969.
[T]Tondeur, Ph.: Affine Zusammenhänge auf Mannigfaltigkeiten mit fastsymplektischer Struktur. Comm. Math. Helv.36, 234–244 (1961).
[TV]Tricerri, F., Vanhecke, L.: Curvature tensors on almost Hermitian manifolds. Trans. Amer. Math. Soc.267, 365–398 (1981).
[W]Weyl, H.: Classical groups, Their Invariants and Representations. Princeton: Princeton Univ. Press. 1946.
[Y]Yau, S. T.: Calabi's conjecture and some new results in algebraic geometry. Proc. Nat. Acad. Sci. USA74, 1798–1799 (1977).
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Vaisman, I. Symplectic curvature tensors. Monatshefte für Mathematik 100, 299–327 (1985). https://doi.org/10.1007/BF01339231
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DOI: https://doi.org/10.1007/BF01339231