4,898 research outputs found
Toric moment mappings and Riemannian structures
Coadjoint orbits for the group SO(6) parametrize Riemannian G-reductions in
six dimensions, and we use this correspondence to interpret symplectic
fibrations between these orbits, and to analyse moment polytopes associated to
the standard Hamiltonian torus action on the coadjoint orbits. The theory is
then applied to describe so-called intrinsic torsion varieties of Riemannian
structures on the Iwasawa manifold.Comment: 25 pages, 14 figures; Geometriae Dedicata 2012, Toric moment mappings
and Riemannian structures, available at
http://www.springerlink.com/content/yn86k22mv18p8ku2
Non-commutative integrable systems on -symplectic manifolds
In this paper we study non-commutative integrable systems on -Poisson
manifolds. One important source of examples (and motivation) of such systems
comes from considering non-commutative systems on manifolds with boundary
having the right asymptotics on the boundary. In this paper we describe this
and other examples and we prove an action-angle theorem for non-commutative
integrable systems on a -symplectic manifold in a neighbourhood of a
Liouville torus inside the critical set of the Poisson structure associated to
the -symplectic structure
Noncommutative geometry and lower dimensional volumes in Riemannian geometry
In this paper we explain how to define "lower dimensional'' volumes of any
compact Riemannian manifold as the integrals of local Riemannian invariants.
For instance we give sense to the area and the length of such a manifold in any
dimension. Our reasoning is motivated by an idea of Connes and involves in an
essential way noncommutative geometry and the analysis of Dirac operators on
spin manifolds. However, the ultimate definitions of the lower dimensional
volumes don't involve noncommutative geometry or spin structures at all.Comment: 12 page
Invariants of pseudogroup actions: Homological methods and Finiteness theorem
We study the equivalence problem of submanifolds with respect to a transitive
pseudogroup action. The corresponding differential invariants are determined
via formal theory and lead to the notions of k-variants and k-covariants, even
in the case of non-integrable pseudogroup. Their calculation is based on the
cohomological machinery: We introduce a complex for covariants, define their
cohomology and prove the finiteness theorem. This implies the well-known
Lie-Tresse theorem about differential invariants. We also generalize this
theorem to the case of pseudogroup action on differential equations.Comment: v2: some remarks and references addee
Equivariant differential characters and symplectic reduction
We describe equivariant differential characters (classifying equivariant
circle bundles with connections), their prequantization, and reduction
Topological constraints on spiral wave dynamics in spherical geometries with inhomogeneous excitability
We analyze the way topological constraints and inhomogeneity in the
excitability influence the dynamics of spiral waves on spheres and punctured
spheres of excitable media. We generalize the definition of an index such that
it characterizes not only each spiral but also each hole in punctured,
oriented, compact, two-dimensional differentiable manifolds and show that the
sum of the indices is conserved and zero. We also show that heterogeneity and
geometry are responsible for the formation of various spiral wave attractors,
in particular, pairs of spirals in which one spiral acts as a source and a
second as a sink -- the latter similar to an antispiral. The results provide a
basis for the analysis of the propagation of waves in heterogeneous excitable
media in physical and biological systems.Comment: 5 pages, 6 Figures, major revisions, accepted for publication in
Phys. Rev.
On the geometry of mixed states and the Fisher information tensor
In this paper, we will review the co-adjoint orbit formulation of finite
dimensional quantum mechanics, and in this framework, we will interpret the
notion of quantum Fisher information index (and metric). Following previous
work of part of the authors, who introduced the definition of Fisher
information tensor, we will show how its antisymmetric part is the pullback of
the natural Kostant-Kirillov-Souriau symplectic form along some natural
diffeomorphism. In order to do this, we will need to understand the symmetric
logarithmic derivative as a proper 1-form, settling the issues about its very
definition and explicit computation. Moreover, the fibration of co-adjoint
orbits, seen as spaces of mixed states, is also discussed.Comment: 27 pages; Accepted Manuscrip
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