A Note on the Moment Map on Symplectic Manifolds
نویسنده
چکیده
We consider a connected symplectic manifold M acted on by a connected Lie group G in a Hamiltonian fashion. If G is compact we study the smooth function f =‖ μ ‖. We prove that if a point x ∈ M realizes a local maximum of the squared moment map ‖ μ ‖ then the orbit Gx is symplectic and Gμ(μ(x)) is G-equivariantly symplectomorphic to a product of a flag manifold and a symplectic manifold which is acted on trivially by G. As an application we characterize completely the symplectic manifolds whose squared moment map is constant. If G is not compact, we characterize the symplectic manifolds acted on by a semisimple Lie group G whose moment map satisfies ‖ μ ‖= 0 and ones whose principal G-orbits are symplectic. These results generalize ones given in [6], [2].
منابع مشابه
Symplectic Toric Manifolds
Foreword These notes cover a short course on symplectic toric manifolds, delivered in six lectures at the summer school on Symplectic Geometry of Integrable Hamiltonian Systems, mostly for graduate students, held at the Centre de Recerca Matemàtica in Barcelona in July of 2001. The goal of this course is to provide a fast elementary introduction to toric manifolds (i.e., smooth toric varieties)...
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