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Nonlinear flag manifolds as coadjoint orbits.

Stefan Haller1, Cornelia Vizman2

  • 1Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.

Annals of Global Analysis and Geometry
|October 22, 2020
PubMed
Summary
This summary is machine-generated.

We introduce nonlinear flags, which are sequences of nested submanifolds, and explore their geometry. This work generalizes nonlinear Grassmannians and describes symplectic nonlinear flags within Hamiltonian diffeomorphism coadjoint orbits.

Keywords:
Coadjoint orbitsFréchet manifoldGroups of diffeomorphismsMoment mapNonlinear GrassmanniansNonlinear flag manifoldsSpaces of embeddings

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Area of Science:

  • Differential Geometry
  • Symplectic Geometry
  • Topology

Background:

  • Nonlinear flags are finite sequences of nested closed submanifolds.
  • Generalizing nonlinear Grassmannians is an active area of research.
  • Understanding coadjoint orbits of Hamiltonian diffeomorphisms is crucial in geometric mechanics.

Purpose of the Study:

  • To study the geometry of Fréchet manifolds of nonlinear flags.
  • To generalize the concept of nonlinear Grassmannians.
  • To describe a class of coadjoint orbits of Hamiltonian diffeomorphisms.

Main Methods:

  • Utilizing the theory of Fréchet manifolds.
  • Applying techniques from differential geometry.
  • Investigating the structure of symplectic submanifolds.

Main Results:

  • The geometry of Fréchet manifolds of nonlinear flags is analyzed.
  • Nonlinear Grassmannians are generalized.
  • A class of coadjoint orbits consisting of nested symplectic submanifolds (symplectic nonlinear flags) is described.

Conclusions:

  • Nonlinear flags provide a new framework for studying geometric structures.
  • The generalization of nonlinear Grassmannians offers new insights into their properties.
  • The identified symplectic nonlinear flags are significant for understanding Hamiltonian dynamics.