Castrillón López, Marco

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First Name
Last Name
Castrillón López
Universidad Complutense de Madrid
Faculty / Institute
Ciencias Matemáticas
Álgebra, Geometría y Topología
Geometría y Topología
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Now showing 1 - 10 of 72
  • Publication
    Distributions admitting a local basis of homogeneous polynomials
    (the American Romanian Academy of Arts and Sciences (USA), 1999) Castrillón López, Marco; Gadea, P.M.; Muñoz Masqué, Jaime
    The paper is a survey of several results by the authors, the main one of them being the following characterization of homogeneous algebraic distributions: Let us consider a vertical distribution D on the vector bundle p:E→M locally spanned by vertical vector fields X1,⋯,Xr. Let χ be the Liouville vector field of the vector bundle. Then there exists an r×r invertible matrix with smooth entries (cij) such that the vector fields Yj=∑ri=1cijXi, 1≤j≤r, are homogeneous algebraic of degree d if and only if there exists an r×r matrix A=(aij) of smooth functions given by [χ,Xj]=∑ri=1aijXi such that A restricted to the zero section of E is(d−1) times the identity matrix. Examples and applications are given.
  • Publication
    Homogeneous quaternionic Kähler structures and quaternionic hyperbolic space
    (Birkhäuser, 2006-12) Castrillón López, Marco; Martínez Gadea, Pedro; Swann, Andrew
    Homogeneous Riemannian structures have been studied and classified in terms of tensors through the works of Ambrose-Singer and of Tricerri-Vanhecke, dating back to the 1950s and 1980s, respectively. More recently, an abstract representation theoretic decomposition of the space V of tensors satisfying the symmetries of a homogeneous Riemannian structure has been proposed by A. Fino [Math. J. Toyama Univ. 21 (1998), 1–22; ] in the context of H-homogeneous structures, H being any of the possible irreducible holonomy groups. The paper under review deals with homogeneous quaternionic Kähler structures and its first result is a concrete description of the decomposition of V into five basic subspaces QK1,…,QK5 invariant under the action of Sp(n)⋅Sp(1), n≥2. Besides this decomposition, the main statements, anticipated in the note [M. Castrillón López, P. M. Gadea and A. Swann, C. R. Math. Acad. Sci. Paris 338 (2004), no. 1, 65–70; ], concern homogeneous quaternionic Kähler structures on the quaternionic hyperbolic space HHn. It is shown in particular that all such structures are in the class QK3 and that they are realized by the homogeneous models Sp(1)RN/Sp(1), where N is the nilpotent factor in the Iwasawa decomposition of Sp(n,1) and the isotropy representation depends on a positive real parameter.
  • Publication
    Poisson–Poincaré reduction for Field Theories
    (2022-10-27) Berbel, M. A.; Castrillón López, Marco
    Given a Hamiltonian system on a fiber bundle, there is a Poisson covariant formulation of the Hamilton equations. When a Lie group G acts freely, properly, preserving the fibers of the bundle and the Hamiltonian density is G-invariant, we study the reduction of this formulation to obtain an analogue of Poisson–Poincaré reduction for field theories. This procedure is related to the Lagrange–Poincaré reduction for field theories via a Legendre transformation. Finally, an application to a model of a charged strand evolving in an electric field is given.
  • Publication
    Elementos de Matemáticas y aplicaciones
    (No publicado, 2013) Castrillón López, Marco; Díaz-Cano Ocaña, Antonio; Etayo Gordejuela, J. Javier; Folgueira, Marta; Infante del Río, Juan Antonio; Pozo Coronado, Luis Miguel; Rey Cabezas, José María
    Material elaborado por profesores de la UCM para la asignatura de este nombre del grado en Ingeniería Matemáticas, el grado en Matemáticas y el grado en Matemáticas y Estadística. Contenidos: - Números enteros. Dígitos de control y criptografía. - Grupos de simetrías. Mosaicos. - Trigonometría plana y esférica. Aplicaciones (Navegación, Astronomía de posición, GPS). - Dinámica discreta. Aplicaciones (Finanzas, introducción al caos). - Teoría de grafos. Algoritmo de ordenación de Google.
  • Publication
    Independence of Yang-Mills Equations with Respect to the Invariant Pairing in the Lie Algebra
    (Springer, 2007) Castrillón López, Marco; Muñoz Masqué, Jaime
    It is proved that the Euler–Lagrange equations of a Yang-Mills type Lagragian is independent with respect to the chosen pairing in the Lie algebra. Moreover, the Hamilton- Cartan equations of these Lagrangians are obtained and proved to be also independent with respect to the pairing.
  • Publication
    Current forms and gauge invariance
    (IOP Publishing, 2004) Castrillón López, Marco; Muñoz Masqué, Jaime
    Let C be the bundle of connections of a principal G-bundle π: P → M,and let V be the vector bundle associated with P by a linear representation G → GL(V ) on a finite-dimensional vector space V . The Lagrangians on J 1(C ×M V) whose current form is gauge invariant, are described and the gauge-invariant Lagrangians on J 1(V) are classified.
  • Publication
    Routh Reduction for Singular Lagrangians
    (Springer-Verlag, 2010) Langerock, Bavo; Castrillón López, Marco
    This paper concerns the Routh reduction procedure for Lagrangians systems with symmetry. It differs from the existing results on geometric Routh reduction in the fact that no regularity conditions on either the Lagrangian L or the momentum map JL are required apart from the momentum being a regular value of JL. The main results of this paper are: the description of a general Routh reduction procedure that preserves the Euler-Lagrange nature of the original system and the presentation of a presymplectic framework for Routh reduced systems. In addition, we provide a detailed description and interpretation of the Euler-Lagrange equa-tions for the reduced system. The proposed procedure includes Lagrangian systems with a non-positively definite kinetic energy metric.
  • Publication
    Gauge interpretation of characteristic classes
    (International Press, 2001) Castrillón López, Marco; Muñoz Masqué, Jaime
    Let π:P→M be a principal G-bundle. Then one can consider the following diagram of fibre bundles: \CD J^{1}(P) @>\pi_{10}>> P\\ @VqVV @VV\pi V\\ C(P) @>p>> M\endCD where p is the bundle of connections of π. As is well known, q is also a principal G-bundle, and the canonical contact form θ on J1(P) can be considered as a connection form on q, with curvature form Θ. One defines aut P as the Lie algebra of G-invariant vector fields on P and gau P as the ideal of π-vertical G-invariant vector fields on P. If X∈autP⊂X(P), then one defines the infinitesimal contact transformation associated to X, X1∈X(J1(P)), and its q-projection XC∈X(C(P)). A differential form Ω on C(P) is said to be aut P-invariant [resp. gauge invariant] if LXCΩ=0 for every X∈autP [resp. X∈gauP]. On the other hand, let us denote by g the Lie algebra of G. An element of the symmetric algebra of g∗ will be called a Weil polynomial. The main result of the paper is the following theorem: If G is connected, for every gauge invariant form Ω on C(P) there exist differential forms ω1,…,ωk on M and Weil polynomials f1,…,fk such that Ω=p∗(ω1)∧f1(Θ)+⋯+p∗(ωk)∧fk(Θ). As a consequence, the authors prove that a differential form Ω on C(P) is aut P-invariant iff Ω=f(Θ), where f is a Weil polynomial, and then Ω is closed. Explicit examples are shown and the link between the above theorem and the geometric formulation of Utiyama's theorem is explained.
  • Publication
    Hamiltonian structure of gauge-invariant variational problems
    (International Press, 2012) Castrillón López, Marco; Muñoz Masqué, Jaime
    Let C→M be the bundle of connections of a principal bundle on M . The solutions to Hamilton–Cartan equations for a gauge-invariant Lagrangian density Λ on C satisfying a weak condition of regularity, are shown to admit an affine fibre-bundle structure over the set of solutions to Euler–Lagrange equations for Λ . This structure is also studied for the Jacobi fields and for the moduli space of extremals.
  • Publication
    Conservation laws as part of Lagrangian reduction. Application to image evolution
    (Springer, 2023-08-01) Castrillón López, Marco; Nielsen, F.; Barbaresco, F.
    This note collects a series of results into a single formulation. Namely, it is proved that, when reduction is performed to a symmetric Lagrangian, the reduced variational equations can be split into two parts, one of them is exactly the Noether theorem. Conservation laws enter into reduction as part of the new variational equations. We give a short description of this situation in the case of evolution of circles in the plane.