Abstract
In this chapter and the chapters that follow, “differentiable” means \(C^{\infty }\)-differentiable, “manifold” means \(C^{\infty }\)-differentiable real manifold, “differential form” and “vector field” mean exterior differential form and \(C^{\infty }\) vector field, respectively.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Added by the authors of the Forewords. Originally, Moser introduced his method for manifolds endowed with a volume form. The use of this method for a proof of Darboux’ theorem is due to the French mathematician Jean Martinet (Sur les singularités de formes différentielles, thèse de doctorat d’État, Grenoble, 1969, and Ann. Inst. Fourier Grenoble, 20, 1970, 95–178). Symplectic coordinates are often called canonical coordinates or Darboux coordinates in other texts.
- 2.
Added by the authors of the Forewords. Symplectic vector fields are often called locally Hamiltonian vector fields in other texts.
- 3.
Added by the authors of the Forewords. The contraction method for the construction of Lagrangian immersions is often called the use of Morse families of functions in other texts. It is due to the Swedish mathematician Lars Hörmander.
- 4.
Added by the authors of the Forewords. The existence of a flat torsionless connection on the leaves of a Lagrangian foliation of a symplectic manifold was observed by the French mathematician Paulette Libermann in her thesis (Sur le problème d’équivalence de certaines structures infinitésimales régulières, thèse de doctorat d’État, Strasbourg, 1953, and Ann. Mat. Pura Appl. 36 , 1954, 27–120).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 Springer Nature Singapore Pte Ltd. and Science Press
About this chapter
Cite this chapter
Koszul, JL., Zou, Y.M. (2019). Symplectic Manifolds. In: Introduction to Symplectic Geometry. Springer, Singapore. https://doi.org/10.1007/978-981-13-3987-5_2
Download citation
DOI: https://doi.org/10.1007/978-981-13-3987-5_2
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-3986-8
Online ISBN: 978-981-13-3987-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)