The Symplectic Normal Space of a Cotangent-Lifted Action

Abstract

For the cotangent bundle of a smooth Riemannian manifold acted upon by the lift of a smooth and proper action by isometries of a Lie group, we characterize the symplectic normal space at any point. We show that this space splits as the direct sum of the cotangent bundle of a linear space and a symplectic linear space coming from reduction of a coadjoint orbit. This characterization of the symplectic normal space can be expressed solely in terms of the group action on the base manifold and the coadjoint representation. Some relevant particular cases are explored.

Replaced with partially rewritten version. Exposition and results are improved and some mistakes fixed

References in corpus (3)

• On the geometry of reduced cotangent bundles at zero momentum
• Singular cotangent bundle reduction and spin Calogero-Moser systems
• Stability of relative equilibria with singular momentum values in simple mechanical systems

Cited by in corpus (6)

• Normal Form of Equivariant Maps and Singular Symplectic Reduction in Infinite Dimensions with Applications to Gauge Field Theory
• Persistence of stationary motion under explicit symmetry breaking perturbation
• Nonlinear Stability of Riemann Ellipsoids with Symmetric Configurations
• The Hamiltonian Tube Of A Cotangent-Lifted Action
• Singular symplectic cotangent bundle reduction of gauge field theory
• Symplectic slice for subgroup actions