Generalized Moser lemma
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Abstract:
We show how the classical Moser lemma from symplectic geometry extends to generalized complex structures (GCS) on arbitrary Courant algebroids. For this, we extend the notion of a Lie derivative to sections of the tensor bundle $(\otimes ^i E)\otimes (\otimes ^j E^*)$ with respect to sections of the Courant algebroid $E$ using the Dorfman bracket. We then give a cohomological interpretation of the existence of one-parameter families of GCS on $E$ and of flows of automorphims of $E$ identifying all GCS of such a family. In the particular case of symplectic manifolds, we recover the results of Moser. Finally, we give a criterion to detect the local triviality of arbitrary GCS which generalizes the Darboux-Weinstein theorem.References
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Additional Information
- Mathieu Stiénon
- Affiliation: Departement Mathematik, E.T.H. Zürich, 8092 Zürich, Switzerland
- Address at time of publication: Institut de Mathématiques de Jussieu, Université Paris-Diderot, 75013 Paris, France
- Email: stienon@math.ethz.ch, stienon@math.jussieu.fr
- Received by editor(s): April 4, 2008
- Published electronically: May 10, 2010
- Additional Notes: This work was supported by the European Union through the FP6 Marie Curie RTN ENIGMA (Contract number MRTN-CT-2004-5652) and by the E.S.I. Vienna through a Junior Research Fellowship.
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 5107-5123
- MSC (2010): Primary 53C15, 17B62, 17B66
- DOI: https://doi.org/10.1090/S0002-9947-10-04965-2
- MathSciNet review: 2657674