Conformally parametrized surfaces associated with CP^(N-1) sigma models

Two-dimensional conformally parametrized surfaces immersed in the su(N) algebra are investigated. The focus is on surfaces parametrized by solutions of the equations for the CP^(N-1) sigma model. The Lie-point symmetries of the CP^(N-1) model are computed for arbitrary N. The Weierstrass formula for immersion is determined and an explicit formula for a moving frame on a surface is constructed. This allows us to determine the structural equations and geometrical properties of surfaces in R^(N^2-1). The fundamental forms, Gaussian and mean curvatures, Willmore functional and topological charge of surfaces are given explicitly in terms of any holomorphic solution of the CP^2 model. The approach is illustrated through several examples, including surfaces immersed in low-dimensional su(N) algebras.Comment: 32 page

Surfaces immersed in su(N+1) Lie algebras obtained from the CP^N sigma models

We study some geometrical aspects of two dimensional orientable surfaces arrising from the study of CP^N sigma models. To this aim we employ an identification of R^(N(N+2)) with the Lie algebra su(N+1) by means of which we construct a generalized Weierstrass formula for immersion of such surfaces. The structural elements of the surface like its moving frame, the Gauss-Weingarten and the Gauss-Codazzi-Ricci equations are expressed in terms of the solution of the CP^N model defining it. Further, the first and second fundamental forms, the Gaussian curvature, the mean curvature vector, the Willmore functional and the topological charge of surfaces are expressed in terms of this solution. We present detailed implementation of these results for surfaces immersed in su(2) and su(3) Lie algebras.Comment: 32 pages, 1 figure; changes: major revision of presentation, clarifications adde

Invariant Forms and Automorphisms of Locally Homogeneous Multisymplectic Manifolds

It is shown that the geometry of locally homogeneous multisymplectic manifolds (that is, smooth manifolds equipped with a closed nondegenerate form of degree > 1, which is locally homogeneous of degree k with respect to a local Euler field) is characterized by their automorphisms. Thus, locally homogeneous multisymplectic manifolds extend the family of classical geometries possessing a similar property: symplectic, volume and contact. The proof of the first result relies on the characterization of invariant differential forms with respect to the graded Lie algebra of infinitesimal automorphisms, and on the study of the local properties of Hamiltonian vector fields on locally multisymplectic manifolds. In particular it is proved that the group of multisymplectic diffeomorphisms acts (strongly locally) transitively on the manifold. It is also shown that the graded Lie algebra of infinitesimal automorphisms of a locally homogeneous multisymplectic manifold characterizes their multisymplectic diffeomorphisms.Comment: 25 p.; LaTeX file. The paper has been partially rewritten. Some terminology has been changed. The proof of some theorems and lemmas have been revised. The title and the abstract are slightly modified. An appendix is added. The bibliography is update

The Tulczyjew triple for classical fields

The geometrical structure known as the Tulczyjew triple has proved to be very useful in describing mechanical systems, even those with singular Lagrangians or subject to constraints. Starting from basic concepts of variational calculus, we construct the Tulczyjew triple for first-order Field Theory. The important feature of our approach is that we do not postulate {\it ad hoc} the ingredients of the theory, but obtain them as unavoidable consequences of the variational calculus. This picture of Field Theory is covariant and complete, containing not only the Lagrangian formalism and Euler-Lagrange equations but also the phase space, the phase dynamics and the Hamiltonian formalism. Since the configuration space turns out to be an affine bundle, we have to use affine geometry, in particular the notion of the affine duality. In our formulation, the two maps $\alpha$ and $\beta$ which constitute the Tulczyjew triple are morphisms of double structures of affine-vector bundles. We discuss also the Legendre transformation, i.e. the transition between the Lagrangian and the Hamiltonian formulation of the first-order field theor

Bifurcation Analysis of Minimizing Harmonic Maps Describing the Equilibrium of Nematic Phases Between Cylinders

Bethuel, F.; Brezis, Haim; Coleman, B.D.; Helein, F.. (1992). Bifurcation Analysis of Minimizing Harmonic Maps Describing the Equilibrium of Nematic Phases Between Cylinders. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/1824