# Lectures on the geometry of Poisson manifolds

@inproceedings{Vaisman1994LecturesOT, title={Lectures on the geometry of Poisson manifolds}, author={Izu Vaisman}, year={1994} }

0 Introduction.- 1 The Poisson bivector and the Schouten-Nijenhuis bracket.- 1.1 The Poisson bivector.- 1.2 The Schouten-Nijenhuis bracket.- 1.3 Coordinate expressions.- 1.4 The Koszul formula and applications.- 1.5 Miscellanea.- 2 The symplectic foliation of a Poisson manifold.- 2.1 General distributions and foliations.- 2.2 Involutivity and integrability.- 2.3 The case of Poisson manifolds.- 3 Examples of Poisson manifolds.- 3.1 Structures on ?n. Lie-Poisson structures.- 3.2 Dirac brackets…

## 954 Citations

### The Geometry of Higher-Order Hamilton Spaces: Applications to Hamiltonian Mechanics

- Mathematics
- 2003

1 Geometry of the k-Tangent Bundle TkM.- 1.1 The Category of k-Accelerations Bundles.- 1.2 Liouville Vector Fields. k-Semisprays.- 1.3 Nonlinear Connections.- 1.4 The Dual Coefficients of a Nonlinear…

### Complementary 2-forms of Poisson structures

- Mathematics
- 1996

Let (M, P) be a Poisson manifold. A 2-form w of M such that the Koszul bracket {03C9, 03C9}P = 0 is called a complementary form of P. Every complementary form yields a new Lie algebroid structure of…

### Poisson cohomology of plane quadratic Poisson structures

- Mathematics
- 1997

As is well-known, Poisson cohomology is of special importance in the theory of Poisson geometry. But unfortunately, the computation is very complicated because of the lack of a powerful method. Let…

### Fe b 20 04 Poisson geometry and Morita equivalence

- Mathematics
- 2004

2 Poisson geometry and some generalizations 3 2.1 Poisson manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Dirac structures . . . . . . . . . . . . . . . . . .…

### A Hodge-type decomposition of holomorphic Poisson cohomology on nilmanifolds

- Mathematics
- 2017

Abstract A cohomology theory associated to a holomorphic Poisson structure is the hypercohomology of a bicomplex where one of the two operators is the classical მ̄-operator, while the other operator…

### On Toric Poisson Structures of Type (1,1) and their Cohomology

- Mathematics
- 2017

We classify real Poisson structures on complex toric manifolds of type $(1,1)$ and initiate an investigation of their Poisson cohomology. For smooth toric varieties, such structures are necessarily…

### ASPECTS OF GEOMETRIC QUANTIZATION THEORY IN POISSON GEOMETRY

- Mathematics
- 2000

This is a survey exposition of the results of (14) on the relationship between the geometric quantization of a Poisson manifold, of its symplectic leaves and its symplectic realizations, and of the…

### M ar 2 01 3 Geometry on Big-Tangent Manifolds by Izu Vaisman

- Mathematics
- 2014

Motivated by generalized geometry, we discuss differential geometric structures on the total space TMof the bundle TM ⊕ T ∗M , where M is a differentiable manifold; TM is called a big-tangent…

### The Geometry of Hamilton and Lagrange Spaces

- Mathematics
- 2001

Preface. 1. The geometry of tangent bundle. 2. Finsler spaces. 3. Lagrange spaces. 4. The geometry of cotangent bundle. 5. Hamilton spaces. 6. Cartan spaces. 7. The duality between Lagrange and…

### D G ] 2 0 A ug 2 00 1 POISSON STRUCTURES ON TANGENT BUNDLES by Gabriel Mitric and Izu Vaisman

- Mathematics
- 2022

The paper starts with an interpretation of the complete lift of a Poisson structure from a manifold M to its tangent bundle TM by means of the Schouten-Nijenhuis bracket of covariant symmetric tensor…