Real analytic structures on a symplectic manifold
Authors:
Frank Kutzschebauch and Frank Loose
Journal:
Proc. Amer. Math. Soc. 128 (2000), 3009-3016
MSC (1991):
Primary 53C15; Secondary 32C05
DOI:
https://doi.org/10.1090/S0002-9939-00-05713-0
Published electronically:
April 28, 2000
MathSciNet review:
1769452
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Abstract | References | Similar Articles | Additional Information
Abstract: We prove that every symplectic manifold possesses a real analytic structure. Moreover this structure is unique up to isomorphism.
- Hans Grauert, On Levi’s problem and the imbedding of real-analytic manifolds, Ann. of Math. (2) 68 (1958), 460–472. MR 98847, DOI https://doi.org/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 https://doi.org/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 https://doi.org/10.2969/aspm/00910041
- Nicolae Mihalache, Special neighbourhoods of subsets in complex spaces, Math. Z. 221 (1996), no. 1, 49–58. MR 1369460, DOI https://doi.org/10.1007/PL00004509
- Jürgen Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286–294. MR 182927, DOI https://doi.org/10.1090/S0002-9947-1965-0182927-5
- Raghavan Narasimhan, Analysis on real and complex manifolds, 2nd ed., Masson & Cie, Éditeur, Paris; North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. Advanced Studies in Pure Mathematics, Vol. 1. 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 https://doi.org/10.1007/BF02565913
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Additional Information
Frank Kutzschebauch
Affiliation:
Mathematisches Institut der Universität, Rheinsprung 21, CH – 4051 Basel, Switzerland
Address at time of publication:
Matematiska Institutionen, Box 480, S-751 06 Uppsala, Sweden
MR Author ID:
330461
Email:
kutzsche@math.uu.se
Frank Loose
Affiliation:
Mathematisches Institut der Universität, Auf der Morgenstelle 10, D – 72076 Tübingen, Germany
Email:
frank.loose@uni-tuebingen.de
Received by editor(s):
December 9, 1998
Published electronically:
April 28, 2000
Additional Notes:
The first author was partially supported by SNF (Schweizerische Nationalfonds)
Communicated by:
Leslie Saper
Article copyright:
© Copyright 2000
American Mathematical Society
