Convexity of Hamiltonian manifolds
Let K be a connected Lie group and M a Hamiltonian
K-manifold. In this paper, we introduce the notion of
convexity of M. It implies that
- the momentum image is convex,
- the moment map has connected fibers, and
- the total moment map is open onto its image.
Conversely, the three properties above imply convexity. We show that
most Hamiltonian manifolds occuring "in nature" are convex (e.g.,
M is compact, complex algebraic, or a cotangent
bundle). Moreover, every Hamiltonian manifold is at least locally
convex.
This is an expanded version of §2 of my paper
Weyl groups of Hamiltonian manifolds, I.
Appeared in: J. Lie Theory 12 (2002) 571-582
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Last updated: March 14, 2004