The convex set of density operators of an N-level quantum mechanical system foliated as a complex flag manifold, where each leaf is identified with the adjoint unitary orbit of the eigenvalues of a density matrix For an isospectral bilinear control system evolving on such an orbit, the state feedback stabilization problem admits a natural Lyapunov-based time-varying feedback design A global description of the domain of attraction of the closed-loop system can be provided based on a ldquoroot-spacerdquo-like structure of the cone of density operators The converging conditions are time independent but depend on the topology of the flag manifold: it is shown that the closed loop must have a number of equilibria at least equal to the Euler characteristic of the manifold, thus imposing topological obstructions to global stabilizability

Feedback Stabilization of Isospectral Control Systems on Complex Flag Manifolds: Application to Quantum Ensembles

The study of Lagrangian submanifolds in K¨ahler manifolds and in the nearly K¨ahler six-sphere has been a very active field over the last quarter of century. In this article we survey the main results done during that period from Riemannian geometric point of view.

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Riemannian geometry of lagrangian submanifolds

Chapter 3 - Riemannian Submanifolds

The low-energy vortex effective actionis constructed in a wide class of systems in a color-flavor locked vacuum, which generalizes the results found earlier in the context of U(N) models. It describes the weak fluctuations of the non-Abelian orientational moduli on the vortex worldsheet. For instance, for the minimum vortex in SO(2N) × U(1) or USp(2N) × U(1)gauge theories, the effective action found is a two-dimensional sigma model living on the Hermitian symmetric spaces SO(2N)/U(N) or USp(2N)/U(N), respectively. The fluctuating moduli have the structure of that of a quantum particle state in spinor representations of the GNO dual ofthe color-flavor SO(2N)C+F or USp(2N)C+F symmetry, i.e. of SO(2N) or of SO(2N + 1). Applied to the benchmark U(N) model our procedure reproduces the known \( \mathbb{C}{P^{N - 1}} \) worldsheet action; our recipe allows us to obtain also the effective vortex action for some higher-winding vortices in U(N) and SO(2N) theories.

Non-Abelian vortex dynamics: effective world-sheet action

Fixed Points of Periodic Differentiable Maps.

Differentiable actions of cyclic groups of odd order on compact, connected, orientable manifolds with only two fixed points are studied in order to obtain conditions under which the representations of the group on the tangent spaces at the fixed points are equivalent. A well-known theorem of Milnor establishes the equivalence of these representations in the case of a semifree action on a homology sphere. A result for more general actions is obtained which yields a generalization of Milnor's theorem for groups of odd order. Introduction. Let E be a smooth action of a cyclic group G, preserving the orientation of a compact, connected, orientable, even-dimensional manifold X and having only two stationary points P and Q. Our objective is to study some conditions under which the representations E(p and E3Q, of the group G, on the tangent spaces to X at P and Q, are equivalent. The kind of conditions that we shall study will involve the G-signature of the G-manifold under consideration. Introduced by Atiyah, Bott, and Singer [1], [2], the G-signature is a powerful tool, which seems especially useful in this type of problem. A revealing example of its power is the following theorem of Milnor which yields an important result about h-cobordant lens spaces [5, (12.12)]. THEOREM ([5, (12.11)], [1, (7.27)]). Let G be a compact group of diffeomorphisms of a homology sphere X, with fixed points P and Q, the action being free except at P and Q. Then the induced representations of G on the tangent spaces Tp and TQ are isomorphic. Along the lines of this theorem, we study actions of a more general nature on compact orientable manifolds; that is, actions not necessarily free outside the fixed point set but with certain restrictions on the fixed point sets of the proper subgroups. The nature of our problem restricts us to cyclic groups of odd order, which makes superfluous the initial condition of "orientation-preserving", since we Received by the editors November 1, 1974. AMS (MOS) subject classqifcations (1970). Primary 57E15, 57E25. ? American Mathematical Society 1976

https://www.ams.org/proc/1976-054-01/S0002-9939-1976-0407871-X/S0002-9939-1976-0407871-X.pdf

Actions of groups of odd order on compact, orientable manifolds

This paper contains a proof of the following property of compact irreducible Hermitian symmetric spaces. If H=G/K where G is a compact simply connected simple Lie group, T a maximal torus of G and F(T,H)=|E1,...,Em is the fixed point set of T on H, then for each pair E i , E j there is a 2-dimentional sphere N ij ⊂ H such that Ei and Ej are antipodal points of Nij.

Spheres in Hermitian Symmetric Spaces and Flag Manifolds

We continue the study of the variety X[M] of planar normal sections on a natural embedding of a flag manifold M. Here we consider those subvarieties of X[M] that are projective spaces. When M=G/T is the manifold of complete flags of a compact simple Lie group G, we obtain our main results. The first one characterizes those subspaces of the tangent space T[T] (M), invariant by the torus action and which give rise to real projective spaces in X[M]. The other one is the following. Let \(\mathfrak{p}\) be the tangent space of the inner symmetric space G/K at [K] . Then RP (\(\mathfrak{p}\)) is maximal in X[M] if and only if π2(G/K) does not vanish.

Maximal Projective Subspaces in the Variety of Planar Normal Sections of a Flag Manifold

In this paper an extension of the 2-number (# 2 (M)) of a symmetric space is given for k-symmetric spaces. The new invariant is computed for flag manifolds which are not symmetric. It turns out to be equal to the Euler-Poincare characteristic

/pdf/the-invariant-of-chen-nagano-on-flag-manifolds-28irfmz9nb.pdf

Cristián U. Sánchez

Papers

Actions of groups of odd order on compact, orientable manifolds

Spheres in Hermitian Symmetric Spaces and Flag Manifolds

Maximal Projective Subspaces in the Variety of Planar Normal Sections of a Flag Manifold

The invariant of Chen-Nagano on flag manifolds

A characterization of extrinsic k-symmetric submanifolds of Rn