Real analytic structures on a symplectic manifold

Kutzschebauch, Frank; Loose, Frank

doi:10.1090/S0002-9939-00-05713-0

Real analytic structures on a symplectic manifold
HTML articles powered by AMS MathViewer

by Frank Kutzschebauch and Frank Loose PDF

Proc. Amer. Math. Soc. 128 (2000), 3009-3016 Request permission

Abstract:

We prove that every symplectic manifold possesses a real analytic structure. Moreover this structure is unique up to isomorphism.

References

Hans Grauert, On Levi’s problem and the imbedding of real-analytic manifolds, Ann. of Math. (2) 68 (1958), 460–472. MR 98847, DOI 10.2307/1970257
Hans Grauert and Reinhold Remmert, Theory of Stein spaces, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 236, Springer-Verlag, Berlin-New York, 1979. Translated from the German by Alan Huckleberry. MR 580152
Victor Guillemin and Shlomo Sternberg, Symplectic techniques in physics, 2nd ed., Cambridge University Press, Cambridge, 1990. MR 1066693
Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original. MR 1336822
Sören Illman, Every proper smooth action of a Lie group is equivalent to a real analytic action: a contribution to Hilbert’s fifth problem, Prospects in topology (Princeton, NJ, 1994) Ann. of Math. Stud., vol. 138, Princeton Univ. Press, Princeton, NJ, 1995, pp. 189–220. MR 1368660
Illman, S. and S. Kankaarinta, Some basic results for real analytic proper $G$-manifolds. Preprint Helsinki 1998
Frank Kutzschebauch, On the uniqueness of the analyticity of a proper $G$-action, Manuscripta Math. 90 (1996), no. 1, 17–22. MR 1387751, DOI 10.1007/BF02568290
——, Eigentliche Wirkungen von Lie-Gruppen auf reell-analytischen Mannigfaltigkeiten. Schriftenreihe des Graduiertenkollegs Geometrie und Mathematische Physik, Ruhr- Universität Bochum, Heft 5 (1994)
Takao Matumoto and Masahiro Shiota, Unique triangulation of the orbit space of a differentiable transformation group and its applications, Homotopy theory and related topics (Kyoto, 1984) Adv. Stud. Pure Math., vol. 9, North-Holland, Amsterdam, 1987, pp. 41–55. MR 896944, DOI 10.2969/aspm/00910041
Nicolae Mihalache, Special neighbourhoods of subsets in complex spaces, Math. Z. 221 (1996), no. 1, 49–58. MR 1369460, DOI 10.1007/PL00004509
Jürgen Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286–294. MR 182927, DOI 10.1090/S0002-9947-1965-0182927-5
Raghavan Narasimhan, Analysis on real and complex manifolds, 2nd ed., Advanced Studies in Pure Mathematics, Vol. 1, Masson & Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. MR 0346855
R. Schoen and S.-T. Yau, Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994. Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu; Translated from the Chinese by Ding and S. Y. Cheng; With a preface translated from the Chinese by Kaising Tso. MR 1333601
Whitney, H., Analytic extensions of differentiable functions defined in closed sets. Trans. Am. Math. Soc. 30 (1934), 63–89
——, Differentiable manifolds. Ann. of Math. 37 (1936), 645–680
H. Whitney and F. Bruhat, Quelques propriétés fondamentales des ensembles analytiques-réels, Comment. Math. Helv. 33 (1959), 132–160 (French). MR 102094, DOI 10.1007/BF02565913