Unit Vector Field Research Articles

Global well-posedness of strong solutions to the two-dimensional inhomogeneous biaxial nematic liquid crystal flow with vacuum

This paper considers the inhomogeneous biaxial nematic liquid crystal flow in a smooth bounded domain Ω⊂R2, where the velocity u and the orthogonal unit vector fields (m,n) admit the Dirichlet and Neumann boundary condition, respectively. By applying piecewise estimate and continuity method, we get the global existence of strong solutions, provided that the basic energy is suitably small. Our result may be regarded as an extension and improvement of Gong-Lin (2022) and Li-Liu-Zhong (2017) to the Neumann boundary condition, where the initial vacuum is allowed. Some new techniques are developed in order to deal with integral estimates caused by the boundary condition, and more complicated model.

Even- and odd-parity stabilities of black holes in Einstein-Aether gravity

In Einstein-Aether theories with a timelike unit vector field, we study the linear stability of static and spherically symmetric black holes against both even- and odd-parity perturbations.For this purpose, we formulate a gauge-invariant black hole perturbation theory in the background Aether-orthogonal frame where the spacelike property of hypersurfaces orthogonal to the timelike Aether field is always maintained even inside the metric horizon. Using a short-wavelength approximation with large radial and angular momenta, we show that, in general, there are three dynamical degreesof freedom arising from the even-parity sector besides two propagating degrees of freedom present in the odd-parity sector. The propagation speeds of even-parity perturbations and their no-ghost conditions coincide with those of tensor, vector, and scalar perturbations on the Minkowski background, while the odd sector contains tensor and vector modes with the same propagation speeds as those in the even-parity sector (and hence as those on the Minkowski background).Thus, the consistent study of black hole perturbations in the Aether-orthogonal frame on static and spherically symmetric backgrounds does not add new small-scale stability conditions to those known for the Minkowski background in the literature.

Lifting Directional Fields to Minimal Sections

Directional fields, including unit vector, line, and cross fields, are essential tools in the geometry processing toolkit. The topology of directional fields is characterized by their singularities. While singularities play an important role in downstream applications such as meshing, existing methods for computing directional fields either require them to be specified in advance, ignore them altogether, or treat them as zeros of a relaxed field. While fields are ill-defined at their singularities, the graphs of directional fields with singularities are well-defined surfaces in a circle bundle. By lifting optimization of fields to optimization over their graphs, we can exploit a natural convex relaxation to a minimal section problem over the space of currents in the bundle. This relaxation treats singularities as first-class citizens, expressing the relationship between fields and singularities as an explicit boundary condition. As curvature frustrates finite element discretization of the bundle, we devise a hybrid spectral method for representing and optimizing minimal sections. Our method supports field optimization on both flat and curved domains and enables more precise control over singularity placement.

Lagrangian dynamics and regularity of the spin Euler equation

We derive the spin Euler equation for ideal flows by applying the spherical Clebsch mapping. This equation is based on the spin vector, a unit vector field encoding vortex lines, instead of the velocity. The spin Euler equation enables a feasible Lagrangian study of fluid dynamics, as the isosurface of a spin-vector component is a vortex surface and material surface in ideal flows. We establish a non-blowup criterion for the spin Euler equation, suggesting that the Laplacian of the spin vector must diverge if the solution forms a singularity at some finite time. The direct numerical simulations (DNS) of three ideal flows – the vortex knot, the vortex link and the modified Taylor–Green flow – are conducted by solving the spin Euler equation. The evolution of the Lagrangian vortex surface illustrates that the regions with large vorticity are rapidly stretched into spiral sheets. The DNS result exhibits a pronounced double-exponential growth of the maximum norm of Laplacian of the spin vector, showing no evidence of the finite-time singularity formation if the double-exponential growth holds at later times. Moreover, the present criterion with Lagrangian nature appears to be more sensitive than the Beale–Kato–Majda criterion in detecting the flows that are incapable of producing finite-time singularities.

Timelike surfaces in [formula omitted] with a canonical null direction

Let N be a Lorentz surface and let ∂t be a unit vector field on the Lorentz manifold N×R tangent to R. A timelike surface Σ in N×R is said to have a canonical null direction with respect to ∂t if the projection ∂t⊤ on the tangent space of Σ gives a lightlike vector field. We prove that these surfaces are minimal and ruled, whose rulings are lightlike geodesics and lines of curvature. Other results include the fact that these surfaces are flat if they are totally geodesic. Furthermore, let us assume that the surface Σ is the graph of a function f:N→R. Then it has a canonical null direction if and only if the gradient of f is a lightlike vector field on N. Another property is that the Gaussian curvature of Σ coincides with the Gaussian curvature of N and the sectional curvature of N×R, along tangent planes to Σ. In particular such surfaces in S12×R have constant curvature. We prove, using isothermal coordinates, that functions on N whose gradient is lightlike locally always exist. We give several examples of such surfaces.

Twisted curve geometry underlying topological invariants

Topological invariants such as winding numbers and linking numbers appear as charges of topological solitons in diverse nonlinear physical systems described by a unit vector field defined on two and three dimensional manifolds. While the Gauss-Bonnet theorem shows that the Euler characteristic (a topological invariant) can be written as the integral of the Gaussian curvature (an intrinsic geometric quantity), the intriguing question of whether winding and linking numbers can also be expressed as integrals of some other kinds of intrinsic geometric quantities has not been addressed in the literature. In this paper we provide the answer by showing that for the winding number in two dimensions, these geometric quantities involve torsions of the two evolving space curves describing the manifold. On the other hand, in three dimensions we find that in addition to torsions, intrinsic twists of the space curves are necessary to obtain a nontrivial winding number and linking number. They arise from the hitherto unknown connections that we establish between these topological invariants and the corresponding appropriately normalized global space curve anholonomies (i.e., geometric phases) that can be associated with the unit vector fields on the respective manifolds. An application of our results to a 3D Heisenberg ferromagnetic model supporting a topological soliton is also presented.

Pseudo-symmetric almost cosymplectic 3-manifolds

In this paper, we study the semi-symmetry and pseudo-symmetry of almost cosymplectic [Formula: see text]-manifolds. First, we prove that an [Formula: see text]-almost cosymplectic [Formula: see text]-manifold [Formula: see text] is semi-symmetric if and only if it is cosymplectic. Here by an [Formula: see text]-almost cosymplectic [Formula: see text]-manifold, we mean an almost cosymplectic [Formula: see text]-manifold whose characteristic vector field [Formula: see text] is a harmonic unit vector field. If an almost cosymplectic [Formula: see text]-manifold [Formula: see text] whose fundamental endomorphism field [Formula: see text] is parallel in the direction of the characteristic vector field [Formula: see text] ([Formula: see text]), then it is [Formula: see text]-almost cosymplectic. In particular, an almost cosymplectic [Formula: see text]-manifold [Formula: see text] satisfying [Formula: see text] is semi-symmetric if and only if it is cosymplectic. Next, we prove that pseudo-symmetric [Formula: see text]-almost cosymplectic [Formula: see text]-manifolds are certain generalized almost cosymplectic [Formula: see text]-spaces.

Geodesic and conformally Reeb vector fields on flat 3-manifolds

A unit vector field on a Riemannian manifold M is called geodesic if all of its integral curves are geodesics. We show, in the case of M being a complete flat 3-manifold not equal to E3, that every such vector field is tangent to a 2-dimensional totally geodesic foliation. Furthermore, it is shown that a geodesic vector field X on a closed orientable complete flat 3-manifold is (up to rescaling) the Reeb vector field of a contact form if and only if there is a contact structure transverse to X that is given as the orthogonal complement of some other geodesic vector field. An explicit description of the lifted contact structures (up to diffeomorphism) on the 3-torus is given in terms of the volume of X. Finally, similar results for non-closed flat 3-manifolds are discussed.

Calibrations for the Volume of Unit Vector Fields in Dimension 2

We use the theory of calibrations to write an equation of a minimal volume vector field on a given Riemann surface.

Minimally Immersed Klein Bottles in the Unit Tangent Bundle of the Unit 2-Sphere Arising from Area-Minimizing Unit Vector Fields on $${\mathbb {S}}^2\backslash \{N,S\}$$

Minimally Immersed Klein Bottles in the Unit Tangent Bundle of the Unit 2-Sphere Arising from Area-Minimizing Unit Vector Fields on $${\mathbb {S}}^2\backslash \{N,S\}$$

Magnetic unit vector fields

We show that a unit vector field on an oriented Riemannian manifold is a critical point of the Landau Hall functional if and only if it is a critical point of the Dirichlet energy functional. Therefore, we provide a characterization for a unit vector field to be a magnetic map into its unit tangent sphere bundle. Then, we classify all magnetic left invariant unit vector fields on $3$-dimensional Lie groups.

Gauss maps of harmonic and minimal great circle fibrations

We investigate Gauss maps associated to great circle fibrations of the euclidean unit 3-sphere mathbb {S}^3. We show that the associated Gauss map to such a fibration is harmonic, respectively minimal, if and only if the unit vector field generating the great circle foliation is harmonic, respectively minimal. These results can be viewed as analogues of the classical theorem of Ruh and Vilms about the harmonicity of the Gauss map of a minimal submanifold in the euclidean space. Moreover, we prove that a harmonic or minimal unit vector field on mathbb {S}^3, whose integral curves are great circles, is a Hopf vector field.

*-Ricci tensor on three dimensional almost coKähler manifolds

In this paper, we obtain some classification results of three-dimensional non-coK?hler almost coK?hler manifold M whose Reeb vector field is strongly normal unit vector field with ?(???h?) = 0, for which the *-Ricci tensor is of Codazzi-type or M satisfies the curvature condition Q* ? R = 0.

Ricci Soliton and Certain Related Metrics on a Three-Dimensional Trans-Sasakian Manifold

In this article, a Ricci soliton and *-conformal Ricci soliton are examined in the framework of trans-Sasakian three-manifold. In the beginning of the paper, it is shown that a three-dimensional trans-Sasakian manifold of type (α,β) admits a Ricci soliton where the covariant derivative of potential vector field V in the direction of unit vector field ξ is orthogonal to ξ. It is also demonstrated that if the structure functions meet α2=β2, then the covariant derivative of V in the direction of ξ is a constant multiple of ξ. Furthermore, the nature of scalar curvature is evolved when the manifold of type (α,β) satisfies *-conformal Ricci soliton, provided α≠0. Finally, an example is presented to verify the findings.

Vector field-based curved layer slicing and path planning for multi-axis printing

• A new vector field-based curved layer slicing and printing path planning method is proposed. • A quantitative optimization model is established to find the optimal filament orientation and printing orientation vector fields. • Multi-factors including support-free requiement, collision-free condition, and mechanical performance are considered. • A continuous printing path planning algorithm is developed to reduce the number of nozzle retractions. The emergence of multi-axis printing systems provides a new solution to print complex parts. However, the current process planning methods fail to balance the support-free requirement, the collision-free condition, as well as the mechanical performance at the same time when printing parts with complicated features. This paper proposes a new curved layer slicing and continuous printing path planning method that considers these factors. The proposed method makes use of two mutually orthogonal unit vector fields embedded on the tetrahedral mesh of the part: the filament orientation vector field and the printing orientation vector field, which indicate the filament orientation and the nozzle orientation at each position during the printing process respectively. A quantitative optimization model is established to compute the two optimal unit vector fields. Then a monotonically increasing scalar distance field is generated by integrating along the optimal printing orientation vector field, and the isosurfaces of the distance field can be used to naturally slice the part into curved layers. Finally, at each isosurface, a continuous printing path is planned along the optimal filament orientation field by interpolating isolines of a surface embedded scalar distance field. Both the simulation and physical experiments were conducted to confirms the effectiveness of the proposed method and its advantage over the existing methods in balance different factors including support-free condition, collision-free requirement, and mechanical performance.

THE NOTES ON ENERGY AND ELECTROMAGNETIC FIELD VECTORS IN THE NULL CONE Q2

In this paper, we characterize the directional derivatives in accordance with the asymptotic orthonormal frame {x,α,y} in Q^2. Also, we express the extended Serret-Frenet relations by using cone Frenet formulas and we explain the geometrical understanding of energy on each asymptotic orthonormal vector fields in null cone. Furthermore, we express the bending elastic energy function for the same particle according to curve x(s,ξ,η) and we finalize our results by providing energy variation sketches according to directional derivatives for different cases. Additionally, we explain a geometrical interpretation of the energy for unit vector fields and we express Maxwell’s equations for the electric and magnetic field vectors in null cone 3-space.

A Robust Convolutional Neural Network for 6D Object Pose Estimation from RGB Image with Distance Regularization Voting Loss

Six-degree (6D) pose estimation of objects is important for robot manipulation but at the same time challenging when dealing with occluded and textureless objects. To overcome this challenge, the proposed method presents an end-to-end robust network for real-time 6D pose estimation of rigid objects using the RGB image. In this proposed method, a fully convolutional network with a features pyramid is developed that effectively boosts the accuracy of pixelwise labeling and direction unit vector field that take part in the voting process for object keypoints estimation. The network further takes into account measuring the distance between pixel and keypoint, which aims to help select accurate hypotheses in the RANSAC process. This avoids hypothesis deviations caused by the errors due to direction unit vectors in cases of distant pixels from keypoints. A vectorial distance regularization loss function is used to help Perspective-n-Point find 2D-3D correspondences between 3D object keypoints and their estimated corresponding 2D counterparts. Experiments are performed on widely used LINEMOD and occlusion LINEMOD datasets with ADD (-S) and 2D projection evaluation metrics. The results show that our method improves pose estimation performance compared to the state-of-the-art while still achieving real-time efficiency.

Continuity Equation and Characteristic Flow for Scalar Hencky Plasticity

AbstractWe investigate uniqueness issues for a continuity equation arising out of the simplest model for plasticity, Hencky plasticity. The associated system is of the form where 𝜇 is a nonnegative measure and 𝜎 a two‐dimensional divergence‐free unit vector field. After establishing the Sobolev regularity of that field, we provide a precise description of all possible geometries of the characteristic flow, as well as of the associated solutions. © 2022 Courant Institute of Mathematics and Wiley Periodicals LLC.

Higher transgressions of the Pfaffian

We define transgressions of arbitrary order, with respect to families of unit-vector fields indexed by a polytope, for the Pfaffian of metric connections for semi-Riemannian metrics on vector bundles. We apply this formula to compute the Euler characteristic of a Riemannian polyhedral manifold, very much in the spirit of Chern’s differential-geometric proof of the generalized Gauss–Bonnet formula on closed manifolds and on manifolds-with-boundary. As a consequence, we derive an identity for spherical and hyperbolic polyhedra linking the volumes of faces of even codimension and the measures of outer angles.

Curvature conditions for spatial isotropy

In the context of mathematical cosmology, the study of necessary and sufficient conditions for a semi-Riemannian manifold to be a (generalized) Robertson-Walker space-time is important. In particular, it is a requirement for the development of initial data to reproduce or approximate the standard cosmological model. Usually these conditions involve the Einstein field equations, which change if one considers alternative theories of gravity or if the coupling matter fields change. Therefore, the derivation of conditions which do not depend on the field equations is an advantage. In this work we present a geometric derivation of such a condition. We require the existence of a unit vector field to distinguish at each point of space two (non-equal) sectional curvatures. This is equivalent for the Riemann tensor to adopt a specific form. Our geometrical approach yields a local isometry between the space and a Robertson-Walker space of the same dimension, curvature and metric tensor sign (the dimension of the largest subspace on which the metric tensor is negative definite). Remarkably, if the space is simply-connected, the isometry is global. Our result generalizes to a class of spaces of non-constant curvature the theorem that spaces of the same constant curvature, dimension and metric tensor sign must be locally isometric. Because we do not make any assumptions regarding field equations, matter fields or metric tensor sign, one can readily use this result to study cosmological models within alternative theories of gravity or with different matter fields.