Compatibility equations adapted to magnetic geometry.
problem No specific problem stated; magnetic geometry is the setting.
method Established compatibility equations in magnetic geometry.
result Analogues of Gauss, Ricci, and Codazzi-Mainardi equations in magnetic geometry.
Study shows finiteness of magnetic hypersurfaces on closed manifolds.
problem Understanding the finiteness of magnetic hypersurfaces on closed manifolds.
method Introduced a dynamical version of the second fundamental form to generalize a previous result.
result Real-analytic negatively s-curved magnetic systems on closed real-analytic manifolds have only finitely many closed totally s-magnetic hypersurfaces. Normal forms prove dynamical results for magnetic fields on surfaces.
problem Existence and rigidity of Zoll flows on surfaces.
method Proved normal forms for strong magnetic fields and used them to derive dynamical results.
result Flow cannot be Zoll unless specific conditions hold.
Study magnetic flows on 3D contact sub-Riemannian manifolds using Rumin complex.
problem Understanding magnetic flows on 3D contact sub-Riemannian manifolds.
method Introducing horizontal magnetic flows via closed Rumin differential two-forms and analyzing the lifted sub-Riemannian structure.
result Horizontal magnetic flows can be interpreted as geodesic flows on a suitably lifted structure, which is of Engel type when the magnetic field is non-vanishing.
Study rigidity of Hamiltonians near a minimum in symplectic and magnetic settings.
problem Rigidity of Hamiltonians near a minimum in symplectic and magnetic settings.
method Analyzing Hamiltonian systems near a compact symplectic Morse-Bott minimum, focusing on Zoll flows and magnetic forms.
result A constant curvature quantity characterizes complex space forms among Kähler manifolds.
Paper studies non-associativity in quantum systems with magnetic fields.
problem Non-associativity of magnetic translations in quantum systems.
method Quantum field theory approach with n-component fermions. result Non-associativity described by a 3-cocycle of Rn with values in S1. Paper introduces magnetic Hodge Laplacian for differential forms.
problem No specific problem stated; general spectral analysis of differential forms.
method Introduced magnetic Hodge Laplacian, discussed spectral results.
result Similarities and differences with magnetic Laplacian on functions.
We consider a compact Riemannian manifold M endowed with a potential 1-form A and study the magnetic Laplacian associated with those data (with Neumann magnetic boundary condition if the bpoundary of M is not empty). We first establish a family of upper bounds for all the eigenvalues, compatible with the Weyl law. When…
Study eigenvalues of magnetic Steklov problem on Riemannian annuli.
problem Eigenvalues of magnetic Steklov problem on Riemannian annuli.
method Sharp upper bounds, maximizers, and existence of maximizers for eigenvalues.
result Existence of maximizers for the second normalized eigenvalue for rotationally invariant metrics.
The study shows that the second fundamental form is intrinsic under certain conditions in space forms.
problem Understanding the intrinsic nature of the second fundamental form in space forms.
method Proving the intrinsic nature of the normalized second fundamental form A under specific conditions. result The normalized second fundamental form A is intrinsic if σ2k+1(A)eq0 for some k≥1. The expression for the variation of the area functional of the second fundamental form of a hypersurface in a Euclidean space involves the so-called "mean curvature of the second fundamental form". Several new characteristic properties of (hyper)spheres, in which the mean curvature of the second fundamental form occurs…
The paper explores how magnetic systems' spectra can identify metrics and 1-forms.
problem Can the marked magnetic action spectrum of magnetic systems with Anosov flow determine the metric and 1-form?
method The paper addresses this question in two settings: locally for systems with close metrics and 1-forms, and for metrics in the same conformal class.
result The paper answers the question affirmatively in both settings.
Proves magnetic geodesic flow on sphere is integrable with constant 2-form.
problem Proving integrability of magnetic geodesic flow on sphere.
method Analyzes magnetic geodesic flow on sphere with constant 2-form.
result Proves Liouville integrability of magnetic geodesic flow on sphere.
Study classifies 3D self-shrinkers with constant second form norm.
problem Classifying self-shrinkers with specific geometric properties.
method Analyzes 3D self-shrinkers in Euclidean space with constant second form norm.
result Classifies complete self-shrinkers with constant norm of the second fundamental form.
Paper introduces magnetic Steklov operator on differential forms and its properties.
problem Analyzing the boundary value problem of magnetic Steklov operator.
method Introduced magnetic Steklov operator and proved its well-posedness. Also, computed spectral properties.
result An analogue of Diamagnetic Inequality does not always hold for magnetic Steklov operators.
Paper classifies special Euclidean hypersurfaces with specific geometric properties.
problem Classifying Euclidean hypersurfaces with semi-parallel Moebius second fundamental form.
method Complete classification of hypersurfaces with three distinct principal curvatures.
result Classification of Euclidean umbilic-free hypersurfaces with semi-parallel Moebius second fundamental form.
This paper studies Hamilton-Jacobi equations for magnetic systems with constraints.
problem Understanding dynamics of magnetic systems with geometric constraints.
method Developed Hamilton-Jacobi equations for magnetic systems with nonholonomic constraints.
result Revealed relationships between magnetic structures, constraints, and dynamics.
Non-degeneracy of critical points proven for manifold's squared norm of second fundamental form.
problem Proving non-degeneracy of critical points for a manifold's squared norm of second fundamental form.
method Generic Riemannian metric and conformal class restriction.
result Squared norm of the second fundamental form is a Morse function with non-degenerate critical points.
Study on immersions with flat normal bundle in curved spaces.
problem Behavior of isometric immersions with negative curvature.
method Investigation of second fundamental form growth in space forms.
result Second fundamental form grows exponentially if normal bundle is flat.
The object of study of this article is compact surfaces in the three-dimensional hyperbolic space with a positive-definite second fundamental form. It is shown that several conditions on the Gaussian curvature of the second fundamental form can be satisfied only by extrinsic spheres.
In this paper we show that a complete and non-compact surface immersed in the Euclidean space with quadratic extrinsic area growth has finite total curvature provided the surface has tamed second fundamental form and admits total curvature. In such a case we obtain as well a generalized Chern-Osserman inequality. In th…
A quasiclassical method approximates magnetic monopole eigenvalues.
problem Describing eigenvalues of magnetic Laplacian in non-exact magnetic fields.
method Applying multidimensional WKB method to nontrivial line bundles.
result Demonstrated for Dirac magnetic monopole on a sphere.
We obtain an infinite family of complete non embedded rotational surfaces in R3 whose second fundamental forms have length equal to one at any point. Also we prove that a complete rotational surface with second fundamental form of constant length is either a round sphere, a circular cylinder or, up to a homo…
An expression for the first variation of the area functional of the second fundamental form is given for a hypersurface in a semi-Riemannian space. The concept of the "mean curvature of the second fundamental form" is then introduced. Some characterisations of extrinsic hyperspheres in terms of this curvature are given…
The article studies eigenvalues and spectrum of magnetic Dirac operators.
problem Fundamental mathematical properties of magnetic Dirac operators remain unexplored.
method Eigenvalue estimates and explicit spectrum computation for specific cases.
result Explicit computation of spectrum for magnetic fields on specific manifolds.
We show that a complete submanifold M with tamed second fundamental form in a complete Riemannian manifold N with sectional curvature KN≤κ≤0 are proper, (compact if N is compact). In addition, if N is Hadamard then M has finite topology. We also show that the fundamental tone is an obstruction fo…
It is an interesting question whether a given equation of motion has a periodic solution or not, and in the positive case to describe them. We investigate periodic magnetic curves in elliptic Sasakian space forms and we obtain a quantization principle for periodic magnetic flowlines on Berger spheres. We give a criteri…
This study explores magnetic trajectories on the Heisenberg group, finding symmetries and solutions.
problem Understanding magnetic geodesics on the Heisenberg group with invariant Lorentz force.
method Analyzing the Heisenberg Lie group with a non-commutative product, deriving magnetic equations, identifying symmetries, and solving variational problems.
result Magnetic trajectories are solutions to a variational problem, providing explicit examples of Lagrangians.
Researchers classify special curved spheres in a complex space.
problem Classifying special holomorphic two-spheres in a complex Grassmannian.
method Completely classified noncongruent spheres with constant curvature and second fundamental form.
result Found all homogeneous spheres with constant curvature and second fundamental form.
The study proves properties of self-shrinkers with bounded curvature.
problem Characterizing self-shrinkers with bounded curvature.
method Analyzing properties of self-shrinkers in Rn+1 with bounded second fundamental form. result Proves that if the squared norm of the second fundamental form is bounded, it must be constant.
Study magnetic geodesics on odd spheres, computing critical energy values.
problem Understanding magnetic geodesics on odd-dimensional spheres.
method Explicit computation and analysis of submanifolds and symmetries.
result Energy values determine magnetic geodesic connectivity on spheres.
Study magnetic Hamiltonian systems with constraints, deriving Hamilton-Jacobi equations.
problem Understanding dynamics of controlled magnetic Hamiltonian systems with constraints.
method Defined CMH system, derived Hamilton-Jacobi equations for different constraints.
result Invariant solutions of Hamilton-Jacobi equations under CMH-equivalence.
In this note we show that Hamiltonian stable minimal Lagrangian submanifolds of projective space need not have parallel second fundamental form.
The paper classifies 3D self-expanders with specific properties.
problem Classifying self-expanders with constant curvature components.
method Complete classification of 3D self-expanders with specific curvature conditions.
result Completely classified 3D self-expanders with constant curvature components.
Extends magnetic flow theory results to higher dimensions.
problem Magnetic flows on manifolds of arbitrary dimension.
method New Pestov identities and adapted Riemannian geometry concepts.
result Established tensor tomography and ergodicity results for magnetic flows.
Extends E. Hopf's theorem to magnetic systems without conjugate points.
problem Proving magnetic curvature non-positive for magnetic systems without conjugate points.
method Using magnetic curvature introduced by the first author, proving magnetic flatness conditions.
result Magnetic flatness is a rigid condition with specific metric and curvature properties.
Let M be a closed oriented surface endowed with a Riemannian metric g and let Ω be a 2-form. We show that the magnetic flow of the pair (g,Ω) has zero asymptotic Maslov index and zero Liouville action if and only g has constant Gaussian curvature, Ω is a constant multiple of the area form of g and the mag…
Combines geometric hydrodynamics with magnetic systems to derive new equations and prove well-posedness.
problem Deriving new equations for magnetic systems and proving their well-posedness.
method Introducing the magnetic Euler-Arnold equation and proving well-posedness for specific equations.
result Local and global well-posedness results for the magnetic Euler-Arnold equation associated with the global quasi-geostrophic equations.
We apply a local systolic-diastolic inequality for contact forms and odd-symplectic forms on three-manifolds to bound the magnetic length of closed curves with prescribed geodesic curvature (also known as magnetic geodesics) on an oriented closed surface. Our results hold when the prescribed curvature is either close t…
Metrics are isometric for certain Anosov magnetic systems.
problem Isometry of metrics for Anosov magnetic systems.
method Conjugacy isotopic to the identity, volume-preserving conjugacy, cohomology class.
result Isometric metrics for conjugate Anosov magnetic systems.
Study magnetic fields on special Lie groups, proving non-existence of certain types.
problem Existence of closed 2-forms with specific properties on non-singular 2-step nilpotent Lie groups.
method Analyzing left-invariant magnetic fields on 2-step nilpotent Lie groups, proving non-existence and existence results.
result Strong obstruction and non-existence of closed 2-forms of type II on non-singular Lie algebras.
The complete local classification and geometric description of n-dimensional submanifolds F with recurrent nonparallel second fundamental form in the spaces of constant curvature M(c) are obtained in this article.
In this paper, we determine all conformal minimal immersions of 2-spheres in complex Grassmann manifold G(2,N;C) with parallel second fundamental form.
Develops analysis of weak immersions with bounded second fundamental forms in critical Sobolev space.
problem Analyzing weak immersions with bounded second fundamental forms in a critical Sobolev space.
method Develops analysis of Lipschitz immersions with bounded second fundamental forms in W2n−1,2 space. result Proves existence of C1 differential structure from weak immersions with bounded second fundamental forms. Researchers classify 3D self-shrinkers in 4D space.
problem Classifying complete 3D self-shrinkers with specific properties in Euclidean space.
method Completely classified 3-dimensional complete self-shrinkers with constant norm of the second fundamental form and constant f3 in R4. result A complete classification of 3D self-shrinkers in Euclidean space R4. Study shows the second fundamental form of pseudospherical surfaces is universal and not dependent on specific solutions.
problem Dependence of the second fundamental form in local isometric immersions of pseudospherical surfaces.
method Analysis of third order differential equations and jets of finite order.
result The second fundamental form of pseudospherical surfaces is universal and not dependent on the specific solution.
A spacelike surface in four-dimensional Lorentz-Minkowski spacetime through the lightcone has a meaningful lightlike normal vector field η. Several sufficient assumptions on such a surface with non-degenerate η-second fundamental form are established to prove that it must be a totally umbilical round sphere. With t…
We consider magnetic flows on 2-step nilmanifolds M=Γ\G, where the Riemannian metric g and the magnetic field σ are left-invariant. Our first result is that when σ represents a rational cohomology class and its restriction to g=TeG vanishes on the derived algebra, then the associated…