arXiv subjects

Mauro Subils

Publications and source records attributed to Mauro Subils.

9 recordsLinked to original sources

arXiv2026

Magnetic equations on the Heisenberg group: symmetries, solutions and the inverse problem of the calculus of variations

The Heisenberg Lie group $H_3$ is modeled on the differentiable structure of $\mathbb{R}^3$ but equipped with another non-commutative product operation. By fixing the usual metric on the Heisenberg Lie group, this work provides a comprehensive overview of the behavior of magnetic geodesics for any invariant Lorentz force. After writing the magnetic equations, we found symmetries that enable the explicit computation of the magnetic trajectories for any homogeneous exact and non-exact magnetic form. Finally we show that these magnetic trajectories are solutions of a variational problem: we present explicit examples of Lagrangians.

math.DG↗

arXiv2024

Closed Magnetic geodesics on Heisenberg nilmanifolds

In this work we study the existence of closed magnetic geodesics on three-dimensional Heisenberg nilmanifolds for every left-invariant Lorentz force. Our first objective is to establish the existence of closed contractible magnetic geodesics on $H_3$. Once the invariant magnetic field is induced to a compact quotient $M=\Lambda \backslash H_3$, we study magnetic geodesics on $M$. Firstly, we determine conditions on a lattice $\Lambda \subset H_3$ to ensure that a given magnetic geodesic projects to a closed curve on $M$. In particular, we prove that for {\it any} energy level below the Ma\~n\'e critical value there always exists a contractible closed magnetic geodesic on the compact manifold $M$. On the other hand, we show that closed magnetic geodesics do not necessarily exist in every homotopy class. Finally, we present examples of compact quotients $\Gamma_k\backslash H_3$ that admit infinitely many closed magnetic trajectories, as well as examples for which no closed non-contractible magnetic trajectories exist for a given left-invariant Lorentz force.

math.DG↗

arXiv2022

Symplectic structures on low dimensional 2-step nilmanifolds

The aim of this work is the study of symplectic structures on 2-step nilmanifolds. We concentrate in the closeness condition, proving that the existence of a closed 2-form of type II is necessary to get a symplectic structure. In low dimensions, this condition is sufficient in most cases.

math.SG↗

arXiv2022

Magnetic fields on non-singular 2-step nilpotent Lie groups

The aim of this work is the study of left-invariant magnetic fields on 2-step nilpotent Lie groups. While the existence of closed 2-forms for which the center is either nondegenerate or in the kernel of the 2-form, is always guaranteed, the existence of closed 2-forms for which the center is isotropic but not in the kernel of the 2-form, is a special situation. These 2-forms are called of type II. We obtain a strong obstruction for the existence on non-singular Lie algebras. Moreover, we prove that the only $H$-type Lie groups admitting closed 2-forms of type II are the real, complex and quaternionic Heisenberg Lie groups of dimension three, six and seven, respectively. We also prove the non-existence of uniform magnetic fields under certain hypotheses. Finally we give a construction of non-singular Lie algebras, proving that in some families of these examples there are no closed 2-form of type II.

math.DG↗

arXiv2022

Magnetic trajectories on 2-step nilmanifolds

The aim of this work is the study of magnetic trajectories on nilmanifolds. The magnetic equation is written and the corresponding solutions are found for a family of invariant Lorentz forces on a 2-step nilpotent Lie group equipped with a left-invariant metric. Some examples are computed in the Heisenberg Lie groups $H_n$ for $n=3,5$, showing differences with the case of exact forms. Interesting magnetic trajectories related to elliptic integrals appear in $H_3$. The question of existence of closed or periodic magnetic trajectories for every energy level on Lie groups or on compact quotients is treated.

math.DG↗

arXiv2019

Explicit fundamental solution for the operator $L+α|T|$ on the Gelfand pair $(\mathbb{H}_{n},U(n))$

By means of the spherical functions associated to the Gelfand pair $(\mathbb{H}_{n},U(n))$ we define the operator $L+α|T|$, where $L$ denotes the Heisenberg sublaplacian and $T$ denotes de central element of the Heisenberg Lie algebra, we establish a notion of fundamental solution and explicitly compute in terms of the Gauss hypergeometric function. For $α<n$ we use the Integral Representation Theorem to obtain a more detailed expression. Finally, we remark that when $α=0$ we recover the fundamental solution for the Heisenberg sublaplacian given by Folland.

math.FA↗

arXiv2017

Parabolic nilradicals of Heisenberg type, II

Every real simple non-compact Lie algebra not isomorphic to $\mathfrak{so}(1,n)$ contains a unique standard parabolic subalgebra whose nilradical is a generalized Heisenberg algebra. Here we discuss the associated parabolic geometries and the riemannian geometry of the harmonic spaces having the former as conformal infinities.

math.DG↗

arXiv2016

Parabolic nilradicals of Heisenberg type

We show that every non-compact simple real Lie algebra not isomorphic to so(n,1) has a unique conjugacy class of parabolic subalgebras whose nilradical is of Heisenberg type, or non-singular, and give some applications.

math.DG↗

arXiv2011

Solvable models for Kodaira surfaces

We consider three families of lattices on the oscillator group $G$, which is an almost nilpotent not completely solvable Lie group, giving rise to coverings $G \to M_{k, 0} \to M_{k, π} \to M_{k, π/2}$ for $k\in \Z$. We show that the corresponding families of four dimensional solvmanifolds are not pairwise diffeomorphic and we compute their cohomology and minimal models. In particular, each manifold $M_{k, 0}$ is diffeomorphic to a Kodaira--Thurston manifold, i.e. a compact quotient $S^1 \times \Heis_3 (\R) /Γ_k$ where $Γ_k$ is a lattice of the real three-dimensional Heisenberg group $\Heis_3 (\R)$. We summarize some geometric aspects of those compact spaces. In particular, we note that any $M_{k, 0}$ provides an example of a solvmanifold whose cohomology does not depend on the Lie algebra only and which admits many symplectic structures that are invariant by the group $\R \times\Heis_3 (\R)$ but not under the oscillator group $G$.

math.DG↗