87 citations · 877 across the 97 of their papers we have counts for
16 papers · 1 filter
Cartan Geometry and Infinite-Dimensional Kempf-Ness Theory
Tobias Diez, Akito Futaki, Tudor Ratiu
We pioneer the development of a rigorous infinite-dimensional framework for the Kempf-Ness theorem, addressing the significant challenge posed by the absence of a complexification…
Norm-squared of the momentum map in infinite dimensions with applications to Kähler geometry and symplectic connections
Tobias Diez, Tudor S. Ratiu
We initiate the study of the norm-squared of the momentum map as a rigorous tool in infinite dimensions. In particular, we calculate the Hessian at a critical point, show that it i…
Group-valued momentum maps for actions of automorphism groups
Tobias Diez, Tudor S. Ratiu
The space of smooth sections of a symplectic fiber bundle carries a natural symplectic structure. We provide a general framework to determine the momentum map for the action of the…
Realizing the Teichmüller space as a symplectic quotient
Tobias Diez, Tudor S. Ratiu
Given a closed surface endowed with a volume form, we equip the space of compatible Riemannian structures with the structure of an infinite-dimensional symplectic manifold. We show…
Dirac optimal reduction
Tudor S. Ratiu, Madeleine Jotz
The purpose of this paper is to generalize the regular Optimal Reduction Theorem to general proper Dirac actions, formulated both in terms of point and orbit reduction. A compariso…
Induced Dirac structure on isotropy type manifolds
Tudor S. Ratiu, Madeleine Jotz
A new method of singular reduction is extended from Poisson to Dirac manifolds. Then it is shown that the Dirac structures on the strata of the quotient coincide with those of the…
Invariant generators for generalized distributions
Madeleine Jotz, Tudor S. Ratiu
The existence of invariant generators for distributions satisfying a compatibility condition with the symmetry algebra is proved.
Singular reduction of Dirac structures
Madeleine Jotz, Tudor S. Ratiu, Jedrzej Sniatycki
The regular reduction of a Dirac manifold acted upon freely and properly by a Lie group is generalized to a nonfree action. For this, several facts about -invariant vector field…
Dirac structures, nonholonomic systems and reduction
Madeleine Jotz, Tudor Ratiu
The reduction of nonholonomic systems is formulated in terms of Dirac reduction. An optimal reduction method for a class of nonholonomic systems is formulated. Several examples are…
The restricted Grassmannian, Banach Lie-Poisson spaces, and coadjoint orbits
Daniel Beltita, Tudor S. Ratiu, Alice Barbara Tumpach
We investigate some basic questions concerning the relationship between the restricted Grassmannian and the theory of Banach Lie-Poisson spaces. By using universal central extensio…
Group actions on chains of Banach manifolds and applications to fluid dynamics
François Gay-Balmaz, Tudor S. Ratiu
This paper presents the theory of non-smooth Lie group actions on chains of Banach manifolds. The rigorous functional analytic spaces are given to deal with quotients of such actio…
The Lie-Poisson structure of the LAE- equation
François Gay-Balmaz, Tudor S. Ratiu
This paper shows that the time map of the averaged Euler equations, with Dirichlet, Neumann, and mixed boundary conditions is canonical relative to a Lie-Poisson bracket constr…
The dynamics around stable and unstable Hamiltonian relative equilibria
Juan-Pablo Ortega, Tudor S. Ratiu
For a symmetric Hamiltonian system, lower bounds for the number of relative equilibria surrounding stable and formally unstable relative equilibria on nearby energy levels are give…
Bifurcation of relative equilibria in mechanical systems with symmetry
Pascal Chossat, Debra Lewis, Juan-Pablo Ortega +1
The relative equilibria of a symmetric Hamiltonian dynamical system are the critical points of the so-called augmented Hamiltonian. The underlying geometric structure of the system…
Reduction in principal fiber bundles: covariant Euler-Poincare equations
Marco Castrillon Lopez, Tudor S. Ratiu, Steve Shkoller
Let be a principal G-bundle, and let be a G-invariant Lagrangian density. We obtain the Euler-Poincare equations for the reduced Lagran…
Curvature of the Virasoro-Bott group
Peter W. Michor, Tudor Ratiu
We consider a natural Riemannian metric on the infinite dimensional manifold of all embeddings from a manifold into a Riemannian manifold, and derive its geodesic equation in the c…