Convexity of Hamiltonian manifolds

TL;DR

This paper introduces a new notion of convexity for Hamiltonian K-manifolds, linking geometric properties like convex momentum images and connected fibers, and shows most natural examples are convex.

Contribution

It defines convexity for Hamiltonian manifolds and proves the equivalence of several geometric properties, expanding understanding of their structure and local convexity.

Findings

Most natural Hamiltonian manifolds are convex.

Convexity implies convex momentum image and connected fibers.

Every Hamiltonian manifold is locally convex.

Abstract

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., if M is compact, complex algebraic, or a cotangent bundle). Moreover, every Hamiltonian manifold is locally convex. This is an expanded version of section 2 of my paper dg-ga/9712010 on Weyl groups of Hamiltonian manifolds.