Abstract
The aim of this chapter is to introduce the basic problems and (soft!) techniques in symplectic geometry by presenting examples—more exactly series of examples— of almost complex and symplectic manifolds: it is obviously easier to understand the classification of symplectic ruled surfaces if you have already heard of Hirzebruch surfaces for instance.
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Bibliography
V.I. Arnold, Mathematical methods of classical mechanics, Graduate Texts in Mathematics, Springer, 1978.
M. Atiyah, Vector bundles over an elliptic curve, Proc. Lond. Math. Soc. 7 (1957), 414–452.
M. Audin, Hamiltoniens périodiques sur les variétés symplectiques compactes de dimension 4, in Géométrie Symplectique et mécanique, Springer Lecture Notes in Math. 1416 (1990).
M. Audin, The topology of torus actions on symplectic manifolds, Progress in Math. 93, Birkhäuser, 1991.
M. Audin, Exemples de variétés presque complexes, l’Ens. Math. 37 (1991), 175–190.
D. Bennequin, Caustique mystique, in Séminaire Bourbaki, Astérisque 133–134 (1986).
D. Bennequin, Topologie symplectique, convexité holomorphe et structures de contact, in Séminaire Bourbaki, Astérisque 189–190 (1990).
T. Delzant, Hamiltoniens périodiques et image convexe de l’application moment, Bull. Soc. Math. France 116 (1988), 315–339.
J. Duistermaat, G. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982), 259–269.
R.E. Gompf, preprint (1992).
P. Griffiths, J. Harris, Principles of algebraic geometry, Wiley-Inter-science, 1978.
M. Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307–347.
M. Gromov, Partial differential relations, Ergebnisse der Math., Springer, 1986.
A. Grothendieck, Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. Math. 79 (1957), 121–138.
F. Hirzebruch, DeUber eine Klasse von einfach-zusammenhäng enden komplexen Mannigfaltigkeiten, Math. Ann. 124 (1951), 77–86.
A. Jaffe, F. Quinn, “Theoretical mathematics”: Towards a cultural synthesis of mathematics and theoretical physics, Bull. Amer. Math. Soc. 29 (1993), 1–13.
J. Lafontaine, Surfaces de Hirzebruch, in Géométrie riemannienne en dimension 4, Séminaire Arthur Besse, Cedic, Paris (1981).
M. Karoubi, K-theory, an introduction, Grundlehren der math. Wissenschaften, Springer, 1978.
A.A. Kirillov, Elements of the theory of representations, Grundlehren der math. Wissenschaften, Springer, 1976.
S. Kobayashi, K. Nomizu, Differential Geometry, vol. 2, Interscience Publishers, 1969.
P. Libermann, C.-M. Marle, Symplectic geometry and analytic mechanics, Math. and its Appl. 35, Reidei, 1987.
A. Lichnerowicz, Théorie globale des connexions et des groupes d’holonomie, Edizioni Cremonese, Roma, 1955.
J. Marsden, A. Weinstein, Reduction of symplectic manifolds with symmetry, Reports in Math. Phys. 5 (1974), 121–130.
D. McDuff, Examples of simply-connected symplectic non-Kähler manifolds, J. Differ. Geom. 20 (1984), 267–277.
D. McDuff, Blowing up and symplectic embeddings in dimension 4, Topology 30 (1991), 409–421.
D. McDuff, The structure of rational and ruled symplectic manifolds, Journal of the Amer. Math. Soc. 3 (1990), 679–712.
A. Newlander, L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math. 65 (1957), 391–404.
M. Spivak, Differential Geometry, vol. 4, Publish or Perish, 1975.
A. Weinstein, Lectures on symplectic manifolds, Regional Conference Series in mathematics, 1977.
A. Weinstein, On the hypotheses of the Rabinowitz periodic orbit theorems, Journal of Diff. Equations 33 (1979), 353–358.
Wu W.T., Sur les classes caractéristiques des structures fibrées sphériques, Actualités scientifiques et industrielles, Hermann, Paris, 1952.
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Audin, M. (1994). Symplectic and almost complex manifolds. In: Audin, M., Lafontaine, J. (eds) Holomorphic Curves in Symplectic Geometry. Progress in Mathematics, vol 117. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8508-9_3
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DOI: https://doi.org/10.1007/978-3-0348-8508-9_3
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