Non-symplectic smooth circle actions on symplectic manifolds (original) (raw)
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For every compact almost complex manifold (M,J) equipped with a J-preserving circle action with isolated fixed points, a simple algebraic identity involving the first Chern class is derived. This enables us to construct an algorithm to obtain linear relations among the isotropy weights at the fixed points. Suppose that M is symplectic and the action is Hamiltonian. If the manifold satisfies an extra "positivity condition" this algorithm determines a family of vector spaces which contain the admissible lattices of weights.
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Action-angle coordinates and KAM theory for singular symplectic manifolds
arXiv (Cornell University), 2022
Part 1. Introduction and preliminaries Chapter 1. Introduction 1.1. Structure and results of this monograph Chapter 2. A primer on singular symplectic manifolds 2.1. b-Poisson manifolds 2.2. On b m-Symplectic manifolds 2.3. Desingularizing b m-Poisson manifolds Chapter 3. A crash course on KAM theory Part 2. Action-angle coordinates and cotangent models Chapter 4. An action-angle theorem for b m-symplectic manifolds 4.1. Basic definitions 4.2. On b m-integrable systems 4.3. Examples of b m-integrable systems 4.4. Looking for a toric action 4.5. Action-angle coordinates on b m-symplectic manifolds Chapter 5. Reformulating the action-angle coordinate via cotangent lifts 5.1. Cotangent lifts and Arnold-Liouville-Mineur in Symplectic Geometry 5.2. The case of b m-symplectic manifolds Part 3. A KAM theorem for b m-symplectic manifolds Chapter 6. A new KAM theorem 6.1. On the structure of the proof 6.2. Technical results 6.3. A KAM theorem on b m-symplectic manifolds Chapter 7. Desingularization of b m-integrable systems Chapter 8. Desingularization of the KAM theorem on b m-symplectic manifolds Chapter 9. Applications to Celestial mechanics 9.1. The Kepler Problem v vi CONTENTS 9.2. The Problem of Two Fixed Centers 9.3. Double Collision and McGehee coordinates 9.4. The restricted three-body problem 9.5. Escape orbits in Celestial mechanics and Fluid dynamics Bibliography 1 By saying that the diffeomorphism is "ǫ-close to the identity" we mean that, for given H, P and r, there is a constant C such that ψ − Id < Cǫ.
One fixed point actions on low-dimensional spheres
Inventiones Mathematicae, 1990
When one studies the symmetry groups of spheres, disks and Euclidean spaces, it is often very fruitful to begin by comparing the properties of linear symmetry groups with those of more general examples. In particular, if the fixed point set is finite, it is natural to ask if the number of fixed points coincides with the number for some linear action (namely, 0 or 2 for actions on spheres and 1 for actions on disks and Euclidean spaces). During the nineteen forties P.A. Smith, D. Montgomery and H. Samelson raised questions about the existence of compact Lie group actions on spheres with one fixed point and on disks and Euclidean spaces with no fixed points (see [Eil], Problem 39, and [MSa], Section 7). Recently such actions have attracted additional interest in connection with regularity conjectures for algebraic group actions on affine n-space (see [PR]). Of course, the questions for various spaces are related. In particular, continuous fixed point free actions on Euclidean spaces correspond bijectively to one fixed point actions on spheres via one point compactification.
SpringerBriefs in Mathematics
In this chapter, we will define Hamiltonian flows, Hamiltonian actions and moment maps. The layout of the chapter is as follows. In Sect. 2.1 we recall the original example of a Hamiltonian flow, namely, Hamilton's equations. In Sect. 2.2, we will start by understanding what Hamiltonian vector fields and Hamiltonian functions are. In Sect. 2.3, we will introduce a bracket on the set of smooth functions on a symplectic manifold which will satisfy the Jacobi identity and will make the former into a Lie algebra. We will see some examples of such vector fields on S 2 and the 2-torus. In the final section (Sect. 2.4), we will define a moment map and will list some conditions which will guarantee the existence of moment maps and other conditions which guarantee their uniqueness.