Abstract
We will be concerned with two open questions in symplectic geometry. The first question is the existence problem for lagrangian foliations. The second question is to know whether an affine manifold can be embedded as leaf of a lagrangian foliation. We prove that every homogeneous symplectic manifold with “closed regular” action of a solvable Lie group has lagrangian foliations. Moreover such manifold (M,ω) has a “bilagrangian linear connection (M,∇) such that ω is parallel with respect to ∇. About the second question, we prove that every connected and simply connected Lie group with left invariant affine structure can be embedded as leaf of a left invariant lagrangian foliation in a symplectic Lie group (G,ω).
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Bibliographie
R. ABRAHAM and J.E. MARSDEN. Foundation of Mechanics. The Benjamin Cummings Publishing Company-London
L. AUSLANDER. The structure of solvmanifolds. Bull. Amer. Math. Soc. 79 (1973) 227–285
L. AUSLANDER and B. KOSTANT. Polarisation and unitary representation of solvable Lie groups Inv. Math. 14 (1977)
BON YAO CHU. Homogeneous symplectic manifolds. Trans. Amer. Math. Soc. 197 (1974)
P. DAZORD. Sur la géométrie des sous-fibrés et des feuilletages lagrangiens. Ann. Scient. ENS 14 (1981) 465–480
D. FRIED W.M. GOLDMAN and M. HIRSCH. Affine manifolds with nilpotent holonomy Comm. Math. Helv. 56 (1981) 487–527
V.W. GUILLEMIN and S. STERNBERG. Geometric Asymptotics Amer. Math. Soc. Survey 17 (1977). providence RI
H. HESS. On a geometric quantization scheme generalizing those of KOSTANT and SOURIAU and CZYZ (Thèse Berlin 1980)
H. HESS. Connection on symplectic manifolds and geometric quantization. Lecture Notes in Mathematics no 836, 153–166
G. HOCHSCHILD. The structure of Lie groups. Holden Day S.P. California
A.A. KIRILOV. Représentation des groupes et Mécanique Ed. Univ. de Moscou (1971)
J.L. KOSZUL. Déformation des variétés localement plates. Ann. Inst. Fourier (1965)
B. KOSTANT. Orbits, symplectis structures and representation theory. Proc. US-JAPAN Seminar of Differential Geometry KYOTO (1965)
B. KOSTANT. Quantization and unitary representation. Prequantization part I. Springer Lecture Notes 170 (1970) 87–208
P. LIBERMANN. Problème d'équivalence et géométrie symplectique. Astérisque 107–108 (1983), 43–68
A. LICHNEROWICZ. Variété de Poisson et feuilletages (Ann. Fac. Sciences Toulouse vol. IV (1982)
J. MILNOR. Fundamental groups of complete affinely flat manifolds Advances in Math. 25 (1977) 178–187
J.M. MORVAN. Quelques invariants topologiques en géométrie symplectique. Ann. Inst. Poincaré vol. 38/4 (1983)
J.M. MORVAN et L. NIGLIO. Classes caractéristiques de couples de sous-fibrés lagrangiens. (preprint)
NGUIFFO BOYOM. Algèbre à associateur symétrique et algèbres de Lie réductives. (Thèse 3ème cycle. Grenoble (1968))
NGUIFFO BOYOM. Affine embeddings of real Lie groups Proc. of London Math. Soc. LNS 26 (1977) 21–39
NGUIFFO BOYOM (en préparation)
D. SIMS and N. WOODHOUSE. Geometric quantization.Lecture Notes in Physics. Springer-Verlag 53 (1976)
J.M. SOURIAU. Structure des systèmes dynamiques Dunod Université, Paris (1970)
A. WEINSTEIN. Symplectic manifolds and their lagrangian submanifolds. Advances in Math. 6 (1971) 329–346
A. WEINSTEIN. Lecture on symplectic manifolds. Conferences board of Mathematical Science AMS no 29 (1977)
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Nguiffo Boyom, M. Varietes symplectiques affines. Manuscripta Math 64, 1–33 (1989). https://doi.org/10.1007/BF01182083
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DOI: https://doi.org/10.1007/BF01182083