I was invited to give lectures at Nankai University in the spring of 1983. This book is based on the lecture notes, translated and written (with minor modifications) by Yi Ming Zou. We hope to introduce symplectic manifold theory to the readers through this introductory book.
The development of analytical mechanics provided the basic concepts of sym-plectic structures. The term symplectic structure is due largely to analytical mechanics. But in this book, the applications of symplectic structure theory to mechanics are not discussed in any detail; and some of the important parts of the theory, especially the application in analysis, are not discussed at all. For those topics, we refer the readers to the references [1, 2, 7, 26]. The emphasis of this book is on the differential properties of manifolds with symplectic structures.
The first chapter of the book discusses the symplectic structures of vector spaces. The second chapter discusses symplectic manifolds and introduces the basic concepts and the basic results to the readers. We prove the existence of symplectic coordinates (Darboux Theorem) as early as possible in Chap. 2, so the readers can see the importance of the formulas we give in later discussions. The connection between the differentiable functions and the infinitesimal automorphisms of the symplectic structure on a symplectic manifold is the foundation of the symplectic manifold theory, and it will be discussed in Sects. 2.4 and 2.5. This chapter ends with some results on the submanifolds, especially the Lagrangian submanifolds, of a symplectic manifold.
The existence of canonical symplectic structures on cotangent bundles clarifies a lot of questions associated with symplectic structures. Chapter 3 introduces the results on cotangent bundles and symplectic vector fields on cotangent bundles.
Chapter 4 discusses symplectic G-spaces, that is, symplectic manifolds with symplectic structures that are invariant under the actions of some Lie group G. For these symplectic manifolds, certain maps we call moment maps provide with us an effective study tool. The study of symplectic G-spaces is a rich topic in symplectic manifold theory, and there are still many problems that deserve further study. The study of the symplectic G-spaces leads us to the study of the dual structures of Lie algebras and the geometry properties of the so-called coadjoint representations. We will discuss these subjects in Sect. 4.3, which will last until Chap. 5. In Chap. 5, we first introduce some general properties of the so-called Poisson manifolds. The concept of Poisson structures is a generalization of the concept of symplectic structures, and it allows us to consider the classical contents from a new viewpoint. Poisson structures start from the concept of contravariant skew-symmetric tensors. In Sect. 5.3, we will give the precise results for the Poisson structures in the dual spaces of Lie algebras.
Chapter 6, the last chapter, is a special chapter. The purpose of this chapter is to introduce the generalizations of the concepts discussed in Chaps. 2 and 3 to supermanifolds. We only discuss (0, w)-dimensional supermanifolds, that is, we only consider differentiable properties, not geometric properties. In this chapter, we mainly describe the basic properties, omit most the proofs. This is because the materials discussed are rather basic; and, we think that these omitted proofs can serve as exercises for the readers.
Finally, we wish to express our sincere thanks to Professor Zhi-da Yan for his help in the process of the translation and writing of this book.

1 Some Algebra Basics 1

1.1 Skew-Symmetric Forms 1

1.2 0rthogonality Defined by a Skew-Symmetric 2-Form 3

1.3 Symplectic Vector Spaces, Symplectic Bases 6

1.4 The Canonical Linear Representation of sl(2, k) in the Algebra of the Skew-Symmetric Forms on a Symplectic Vector Space 8

1.5 Symplectic Groups 11

1.6 Symplectic Complex Structures 16

2 Symplectic Manifolds 21

2.1 Symplectic Structures on Manifolds 21

2.2 0perators of the Algebra of Differential Forms on a Symplectic

2.3 Symplectic Coordinates 30

2.4 Hamiltonian Vector Fields and Symplectic Vector Fields 35

2.5 Poisson Brackets Under Symplectic Coordinates 44

2.6 Submanifolds of Symplectic Manifolds 48

3 Cotangent Bundles 57

3.1 Liouville Forms and Canonical Symplectic Structures on Cotangent Bundles 57

3.2 Symplectic Vector Fields on a Cotangent Bundle 61

3.3 Lagrangian Submanifolds of a Cotangent Bundle 68

4 Symplectic G-Spaces 75

4.1 Definitions and Examples 76

4.2 Hamiltonian q-Spaces and Moment Maps 79

4.3 Equivariance of Moment Maps 87

5 Poisson Marufolds 91

5.1 The Structure of a Poisson Manifold 91

      5.1.1 The Schouten-Nijenhuis Bracket 91

5.2 The Leaves of a Poisson Manifold 95

5.3 Poisson Structures on the Dual of a Lie Algebra 98

6 A Graded Case 109

6.1 (0, n)-Dimensional Supermanifolds 109

6.2 (0, n)-Dimensional Symplectic Supermanifolds 114

6.3 The Canonical Symplectic Structure on T*P 115

Bibliography 117

Index 119

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