Riemannian moment map

Toric moment mappings and Riemannian structures

We give an interpretation of the symplectic fibrations of coadjoint orbits for the group SU(4) in terms of the compatibility of Riemannian G -reductions in six dimensions. An analysis of moment polytopes associated to a standard Hamiltonian toric action on the coadjoint orbits highlights these relations. This theory leads to a geometrical application to intrinsic torsion classes of 6-dimensiona...

The Moment Map Revisited

In this paper, we show that the notion of moment map for the Hamiltonian action of a Lie group on a symplectic manifold is a special case of a much more general notion. In particular, we show that one can associate a moment map to a family of Hamiltonian symplectomorphisms, and we prove that its image is characterized, as in the classical case, by a generalized “energy-period” relation.

The Moment Map

Projectivity of Moment Map Quotients

Let G be a complex reductive group acting algebraically on a complex projective variety X . Given a polarisation of X , i.e., an ample G-line bundle L over X , Mumford (see [M-F-K]) defined the notion of stability: A point x ∈ X is said to be semistable with respect to L if and only if there exits m ∈ N and an invariant section s : X → L such that s(x) 6= 0. Let X(L) denote the set of semistabl...

Riemannian Geometry: Some Examples, including Map Projections

Some standard formulas are collected on curvature in Riemannian geometry, using coordinates. (There are, as far as I know, no new results). Many examples are given, in particular for manifolds with constant curvature, including many well-known map projections of the sphere and several well-known representations of hyperbolic space, but also some lesser known.