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We extend known constructions of almost-Poisson brackets and their gauge transformations to nonholonomic systems whose Lagrangian is not mechanical but possesses a gyroscopic term linear in the velocities. The new feature introduced by such a term is that the Legendre transformation is an affine, instead of linear, bundle isomorphism between the tangent and cotangent bundles of the configuration space and some care is needed in the development of the geometric formalism. At the end of the day, the affine nature of the Legendre transform is reflected in the affine dependence of the brackets that we construct on the momentum variables. Our study is motivated by a wide class of nonholonomic systems involving rigid bodies with internal rotors which are of interest in control. Our construction provides a natural geometric framework for the (known) Hamiltonisations of the gyrostatic generalisations of the Suslov and Chaplygin sphere problems. [ABSTRACT FROM AUTHOR]

We study the action of Bogoliubov transformations on admissible generalized one-particle density matrices arising in Hartree–Fock–Bogoliubov theory. We show that the orbits of this action are reductive homogeneous spaces, and we give several equivalences that characterize when they are embedded submanifolds of natural ambient spaces. We use Lie theoretic arguments to prove that these orbits admit an invariant symplectic form. If, in addition, the operators in the orbits have finite spectrum, then we obtain that the orbits are actually Kähler homogeneous spaces. [ABSTRACT FROM AUTHOR]

We provide conditions for a compact gradient hyperbolic Ricci and a compact gradient hyperbolic Yamabe soliton to be trivial, hence, the manifold to be an Einstein manifold in the first case, and a manifold of constant scalar curvature, in the second case. In particular, we prove that for a compact gradient hyperbolic Yamabe soliton of dimension > 2 , if the second Lie derivative of the metric in the direction of the potential vector field is trace-free and divergence-free, then the above conclusion is reached. [ABSTRACT FROM AUTHOR]

This article investigates generalized Ricci almost solitons, also known as GRA solitons, on contact metric manifolds, including the gradient case. At first, we establish that a complete K-contact or Sasakian manifold endowed with a closed GRA soliton satisfying 4 c 1 c 2 ≠ 1 is compact Einstein with scalar curvature 2 n (2 n + 1) . As for the gradient case, it exhibits an isometry to the unit sphere S 2 n + 1 . Subsequently, we identify a few adequate conditions under which a non-trivial complete K-contact manifold with a GRA soliton is trivial (η -Einstein). Following that, we establish certain results on H-contact and complete contact manifolds. We also demonstrate that a non-Sasakian (k , μ) -contact manifold with a closed GRA soliton is flat for dimension 3, and for higher dimensions, it is locally isometric to the trivial bundle R n + 1 × S n (4) , provided 4 c 1 c 2 (1 - 2 n) ≠ 1 and c 2 ≠ 0 . Finally, we discuss a few applications of GRA solitons in general relativity. These include characterizing PF spacetimes with a concircular velocity vector field and determining a sufficient condition for a GRW spacetime to be a PF spacetime. [ABSTRACT FROM AUTHOR]

We consider the problem of producing a ruled Möbius band by subjecting an unstretchable, homogeneous, isotropic, elastic material surface material surface in a circular helicoidal reference configuration to a deformation that is isometric and chirality preserving. We find that such a Möbius band is completely determined by the unit binormal of the Frenet frame of its midline, which must be a geodesic and must have uniform torsion inversely proportional to the pitch of the helicoidal reference configuration. Granted that the energy density of the material surface depends quadratically on the mean curvature of its deformed configuration, we show that the total energy stored in producing a ruled Möbius band as described reduces, in closed form and without approximation, to an integral over the midline of the Möbius band. We formulate and numerically solve a constrained variational problem for finding relative minima of the dimensionally reduced bending energy and construct corresponding stable Möbius bands. The only input parameter entering our variational problem is the number ν of turns in a helicoidal reference configuration. We only find solutions if ν exceeds a certain threshold, which we compute to machine precision. Above that threshold, an interplay between the operative constraints leads to a multiplicity of coexisting stable solutions with n ≥ 3 half twists. For each n ≥ 3 , we construct an energetically optimal Möbius band which exhibits n -fold rotational symmetry. All other energy minima yield Möbius bands which lack symmetry. To our knowledge, this study contains the first examples of stable Möbius bands produced by isometrically deforming reference configurations that are not flat. [ABSTRACT FROM AUTHOR]

We conduct a thorough exploration of the magnetic flows and physical dynamics exhibited by point charged particles in three-dimensional ordinary space. Our methodology is grounded in the application of conformable methods. Initially, we refine the general treatment to derive velocity and acceleration vector fields for point particles traversing conformable curves under the influence of distinct force fields. Subsequently, our inquiry extends into the realm of magnetism, where we investigate various types of dynamical magnetic curves that emerge from characterizing the movement of charged particles along conformable curves. In addition to this, we introduce the Fermi–Walker conformable derivative and provide an alternative representation for a unit-speed conformable curve by leveraging the Fermi–Walker connection. Throughout the entirety of the paper, we establish a meaningful correlation between the consistent motion of conformable charged particles and the trajectories dictated by specific external forces within the α -Frenet–Serret frame. The application of fractional calculus plays a pivotal role in our investigations, a facet we meticulously delve into in the paper. This work significantly contributes to an enhanced comprehension of magnetic phenomena and dynamics, offering valuable insights into the behavior of charged particles within conformable frameworks. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the parallel surfaces of the nonlightlike solution of vortex filament equations, namely, the parallel surfaces to Hasimoto surfaces in E 1 3 . We examine the relation between the geometric structures of Hasimoto surfaces and parallel surfaces to them. Moreover, we give concluding remarks about the parameter curves of parallel surface to Hasimoto surfaces. [ABSTRACT FROM AUTHOR]

We recast the physics discussions in the paper of Van den Bleeken (J High Energy Phys 2012(2):67, 2012) within the context of wall-crossing structure à la Kontsevich and Soibelman (Homol Mirror Symmetry Trop Geom, 2014. ). In particular, we compare the Hesse flow given in Van den Bleeken (J High Energy Phys 2012(2):67, 2012) and the attractor flow on the base of the complex integrable system, and show that both can be used in the formalism of wall-crossing structure. We also propose the notions of dual Hesse flow and dual attractor flow, and show that under the rotation of the Z -affine structure, the Hesse flow can be transformed into the dual attractor flow, while the attractor flow into the dual Hesse flow. This suggests its possible use in Mirror Symmetry. [ABSTRACT FROM AUTHOR]

The present study introduces an innovative link between integrable equations and the motion of timelike curves within a three-dimensional Minkowski space. This study aims to establish an anology between the modified generalizations of the Heisenberg spin chain model equation, a complex Korteweg–de Vries equation, and the Ablowitz–Kaup–Newell–Segur hierarchy systems of real type, respectively. This is accomplished through the application of specific functions, which are derived based on the curvatures and torsions of three distinct curves and their corresponding Frenet frames in a 3-dimensional Minkowski space. Making use of this method, the geometric derivation of the integrable equation has been demonstrated with success. [ABSTRACT FROM AUTHOR]

We investigate the Lorentzian rigged structure on null hypersurfaces of a Lorentzian manifold and obtain the necessary and sufficient conditions for such a structure to represent a perfect fluid spacetime, with special attention given to the case of generalized Robertson–Walker spacetimes. [ABSTRACT FROM AUTHOR]

In the current work, we study the geometry and topology of warped product Legendrian submanifolds in Kenmotsu space forms F 2 n + 1 (ϵ) and derive the first Chen inequality, including extrinsic invariants such as the mean curvature and the length of the warping functions. Additionally, sectional curvature and the δ-invariant are intrinsic invariants related to this inequality. An integral bound is also given in terms of the gradient Ricci curvature for the Bochner operator formula of compact warped product submanifolds. Our primary technique is applying geometry to number structures and solving problems such as problems with Dirichlet eigenvalues. [ABSTRACT FROM AUTHOR]

As part of a programme to classify quasi-Einstein metrics (M, g, X) on closed manifolds and near-horizon geometries of extreme black holes, we study such spaces when the vector field X is divergence-free but not identically zero. This condition is satisfied by left-invariant quasi-Einstein metrics on compact homogeneous spaces (including the near-horizon geometry of an extreme Myers–Perry black hole with equal angular momenta in two distinct planes) and on certain bundles over Kähler–Einstein manifolds. We find that these spaces exhibit a mild form of rigidity: they always admit a one-parameter group of isometries generated by X. Further geometrical and topological restrictions are also obtained. [ABSTRACT FROM AUTHOR]

The present paper focused on Vaidya spacetime admitting a conformal Ricci soliton. We derive the expressions for the four components of the associated vector field V and demonstrate that, under this condition, the spacetime reduces to the Schwarzschild spacetime. Furthermore, we determine the necessary expression for the potential function f to establish the existence of a conformal gradient Ricci soliton in the Vaidya spacetime and we provide the condition for which the soliton is shrinking, steady or expanding. [ABSTRACT FROM AUTHOR]

Geometric calculus is a special case of the Non-Newtonian Calculus introduced by Grossman and Katz (Non-Newtonian calculus, Lee Press, Pigeon Cove, 1972). Also, it is a more convenient calculation method for situations where the geometric increment is more meaningful than the arithmetic increment. In this study, geometric curves are defined and geometric Frenet–Serret formulas for these curves are presented. Furthermore, we give applications of these concepts to geometric magnetic curves. [ABSTRACT FROM AUTHOR]

Our purpose in this paper is to study some geometric properties of spacelike hypersurfaces immersed into a pp-wave spacetime, namely, a connected Lorentzian manifold admitting a parallel lightlike vector field. Initially, by applying a new form of maximum principle for smooth functions on a complete noncompact Riemannian manifold, we obtain sufficient conditions which guarantee that a complete noncompact spacelike hypersurface with polynomial volume growth is either totally geodesic, maximal or 1-maximal. As a consequence, we establish nonexistence results concerning such spacelike hypersurfaces. Next, using a weak form of the Omori–Yau maximum principle, we get uniqueness and nonexistence results for stochastically complete spacelike hypersurface with constant mean curvature. Finally, we establish the notion of spacelike mean curvature flow soliton in pp-wave spacetimes and we provide some geometric conditions that allow us to guarantee how close a complete spacelike mean curvature flow soliton is to a totally geodesic immersion. [ABSTRACT FROM AUTHOR]

The classical symmetry method is often employed to find precise solutions to differential equations. This method has yielded several new symmetry reductions and exact solutions for numerous theoretically and physically relevant partial differential equations. These results, as well as the symmetries of a variety of specific cases of the Fokker–Planck equation, were presented in this study using the classical Lie symmetry approach. New exact solutions to the Fokker–Planck equations are provided for each of the six cases. [ABSTRACT FROM AUTHOR]

In this paper, we formulate the theory of nonlinear elasticity in a geometrically intrinsic manner using exterior calculus and bundle-valued differential forms. We represent kinematics variables, such as velocity and rate of strain, as intensive vector-valued forms, while kinetics variables, such as stress and momentum, as extensive covector-valued pseudo-forms. We treat the spatial, material and convective representations of the motion and show how to geometrically convert from one representation to the other. Furthermore, we show the equivalence of our exterior calculus formulation to standard formulations in the literature based on tensor calculus. In addition, we highlight two types of structures underlying the theory: first, the principal bundle structure relating the space of embeddings to the space of Riemannian metrics on the body and how the latter represents an intrinsic space of deformations and second, the de Rham complex structure relating the spaces of bundle-valued forms to each other. [ABSTRACT FROM AUTHOR]

In this paper, a geometric field theory of dislocation dynamics and finite plasticity in single crystals is formulated. Starting from the multiplicative decomposition of the deformation gradient into elastic and plastic parts, we use Cartan’s moving frames to describe the distorted lattice structure via differential 1-forms. In this theory, the primary fields are the dislocation fields, defined as a collection of differential 2-forms. The defect content of the lattice structure is then determined by the superposition of the dislocation fields. All these differential forms constitute the internal variables of the system. The evolution equations for the internal variables are derived starting from the kinematics of the dislocation 2-forms, which is expressed using the notions of flow and of Lie derivative. This is then coupled with the rate of change of the lattice structure through Orowan’s equation. The governing equations are derived using a two-potential approach to a variational principle of the Lagrange–d’Alembert type. As in the nonlinear setting the lattice structure evolves in time, the dynamics of dislocations on slip systems is formulated by enforcing some constraints in the variational principle. Using the Lagrange multipliers associated with these constraints, one obtains the forces that the lattice exerts on the dislocation fields in order to keep them gliding on some given crystallographic planes. Moreover, the geometric formulation allows one to investigate the integrability—and hence the existence—of glide surfaces, and how the glide motion is affected by it. Lastly, a linear theory for small dislocation densities is derived, allowing one to identify the nonlinear effects that do not appear in the linearized setting. [ABSTRACT FROM AUTHOR]

We consider curves γ : [ 0 , 1 ] → R 3 endowed with an adapted orthonormal frame r : [ 0 , 1 ] → S O (3) . We wish to deform such framed curves (γ , r) while preserving two contraints: a local constraint prescribing one of its ‘curvatures’ (i.e., off-diagonal elements of r ′ r T ), and a global constraint prescribing the initial and terminal values of γ and r. We proceed in two stages. First we deform the frame r in a way that is naturally compatible with the constraints on r, by interpreting the local constraint in terms of parallel transport on the sphere. This provides a link to the differential geometry of surfaces. The deformation of the base curve γ is achieved in a second step, by means of a suitable reparametrization of the frame. We illustrate this deformation procedure by providing some applications: first, we characterize the boundary conditions on (γ , r) that are accessible without violating the local constraint; then, we provide a short proof of a smooth approximation result for framed curves satisfying both the differential and the global constraints. Finally, we also apply these ideas to elastic ribbons with nonzero width. [ABSTRACT FROM AUTHOR]

Generalized span categories provide a framework for formalizing mathematicalmodels of open systems in classical mechanics. We introduce categories$\mathsf{LagSy}$ and $\mathsf{HamSy}$ that respectively provide a categoricalframework for the Lagrangian and Hamiltonian descriptions of open classicalmechanical systems. The morphisms of $\mathsf{LagSy}$ and $\mathsf{HamSy}$correspond to such open systems, and composition of morphisms models theconstruction of systems from subsystems. The Legendre transformation gives riseto a functor from $\mathsf{LagSy}$ to $\mathsf{HamSy}$ that translates from theLagrangian to the Hamiltonian perspective.

Contact geometry allows us to describe some thermodynamic and dissipative systems. In this paper we introduce a new geometric structure in order to describe time-dependent contact systems: cocontact manifolds. Within this setting we develop the Hamiltonian and Lagrangian formalisms, both in the regular and singular cases. In the singular case, we present a constraint algorithm aiming to find a submanifold where solutions exist. As a particular case we study contact systems with holonomic time-dependent constraints. Some regular and singular examples are analyzed, along with numerical simulations. [ABSTRACT FROM AUTHOR]

We establish a Penrose-type inequality with angular momentum for four-dimensional, biaxially symmetric, maximal, asymptotically flat initial datasets (M, g, k) for the Einstein equations with fixed angular momenta and horizon inner boundary associated to a 3-sphere outermost minimal surface. Moreover, equality holds if and only if the initial dataset is isometric to a canonical time slice of a stationary Myers-Perry black hole. [ABSTRACT FROM AUTHOR]

The geometry and interactions between the constituents of a liquid crystal, which are responsible for inducing the partial order in the fluid, may locally favor an attempted phase that could not be realized in R 3 . While states that are incompatible with the geometry of R 3 were identified more than 50 years ago, the collection of compatible states remained poorly understood and not well characterized. Recently, the compatibility conditions for three-dimensional director fields were derived using the method of moving frames. These compatibility conditions take the form of six differential relations in five scalar fields locally characterizing the director field. In this work, we rederive these equations using a more transparent approach employing vector calculus. We then use these equations to characterize a wide collection of compatible phases. [ABSTRACT FROM AUTHOR]

We introduce and study the Chaplygin systems with gyroscopic forces. This natural class of nonholonomic systems has not been treated before. We put a special emphasis on the important subclass of such systems with magnetic forces. The existence of an invariant measure and the problem of Hamiltonization are studied, both within the Lagrangian and the almost-Hamiltonian framework. In addition, we introduce problems of rolling of a ball with the gyroscope without slipping and twisting over a plane and over a sphere in R n as examples of gyroscopic SO(n)-Chaplygin systems. We describe an invariant measure and provide examples of S O (n - 2) -symmetric systems (ball with gyroscope) that allow the Chaplygin Hamiltonization. In the case of additional SO(2)-symmetry, we prove that the obtained magnetic geodesic flows on the sphere S n - 1 are integrable. In particular, we introduce the generalized Demchenko case in R n , where the inertia operator of the system is proportional to the identity operator. The reduced systems are automatically Hamiltonian and represent the magnetic geodesic flows on the spheres S n - 1 endowed with the round-sphere metric, under the influence of a homogeneous magnetic field. The magnetic geodesic flow problem on the two-dimensional sphere is well known, but for n > 3 was not studied before. We perform explicit integrations in elliptic functions of the systems for n = 3 and n = 4 and provide the case study of the solutions in both situations. [ABSTRACT FROM AUTHOR]

Using fractional differential geometry, we define different types of conformable magnetic curves to discuss some geometric features of the motion of the charged particle when its geometry is induced by the conformable fractional derivative. We obtain the non-stretching motions and the inextensibility conditions of the conformable curves to focus on their evolutions. Then, as a special case, we obtain the time evolution of the Frenet–Serret vectors of the conformable curves with respect to the optical ferromagnetic and antiferromagnetic cases. Finally, we focus on the time evolution of conformable curves under the case of the optical ferromagnetic and antiferromagnetic conditions to investigate how the anholonomy density is associated with the magnetic flux flow. [ABSTRACT FROM AUTHOR]

The present article deals with static perfect fluid spacetimes on generalized Robertson-Walker spacetimes. At first, we show that a generalized Robertson-Walker spacetime with constant scalar curvature as the spatial factor of a static perfect fluid spacetime becomes a perfect fluid spacetime. Next, we prove that under certain restrictions a generalized Robertson-Walker spacetime turns into an Einstein spacetime. As a consequence it is shown that such a spacetime is of Petrov type I, D or O and in case of 4-dimension, the spacetime turns into a Robertson-Walker spacetime. [ABSTRACT FROM AUTHOR]

The paper is devoted to explore the trajectories of charged particles varying under the influence of Lorentz force of Killing magnetic fields (i.e., closed 2-form corresponding to Killing vector fields) in Egorov 3-spaces. Predominantly, we characterize Killing vector fields and hence determine magnetic trajectories in some Killing magnetic fields on Egorov 3-spaces. We also obtain magnetic trajectories in some Killing magnetic backgrounds on flat Egorov 3-spaces. [ABSTRACT FROM AUTHOR]

In our paper, we computed some new characterizations due to an alternative modified frame N ⃗ , C ⃗ , W ⃗ in Euclidean 3-space and we get general differential equation characterizations of a space curve due to the vectors N ⃗ , C ⃗ , W ⃗ . Furthermore, we investigated some differential equations characterizations of the harmonic and harmonic 1-type curves. [ABSTRACT FROM AUTHOR]

The article is aimed to explore trajectories of charged particles moving under the effect of Lorentz force of Killing magnetic fields (i.e., closed 2-forms corresponding to Killing vector fields) in 3D pp-waves. First, we obtain magnetic trajectories in trivial magnetic background which are nothing but the geodesics of 3D pp-waves. Next, we characterize Killing vector fields and hence explore associated Killing magnetic trajectories in 3D pp-waves. In particular, we explicitly determine Killing magnetic trajectories in 3D Cahen–Wallach space and provide some pictorial examples. [ABSTRACT FROM AUTHOR]