Unit Vector Field Research Articles

Anholonomik Koordinatlar Boyunca Yönlü Enerji Fonksiyonelleri

In this paper, a special case of directional energy functional is investigated by computing the directional energy and pseudoangle of unit vector fields in the ordinary three-dimensional space. This approach is also extended simultaneously to define the critical points of the directional energy functionals of the velocity fields. Then, the restriction of the harmonic maps and the extrema of the directional energy functionals is considered, Finally, we compute directional harmonic and biharmonic equations of the curvature vector fields to generalize total bending or energy of vector fields.

Volume minimising unit vector fields on three dimensional space forms of positive curvature

We prove that on a closed 3-manifold of constant positive curvature the unit vector fields of minimum volume are exactly the Hopf vector fields.

On the Biharmonicity of Vector Fields and Unit Vector Fields

Let (M, g) be a compact Riemannian manifold. Equipping its tangent bundle TM (resp. unit tangent bundle \(T_1M\)) with a pseudo-Riemannian g-natural metric G (resp. \({\tilde{G}}\)), we study the biharmonicity of vector fields (resp. unit vector fields) as maps \((M,g) \rightarrow (TM,G)\) (resp. \((M,g) \rightarrow (T_1M,{\tilde{G}})\)), as well as critical points of the bienergy functional restricted to the set \({\mathfrak {X}}(M)\) (resp. \({\mathfrak {X}}^1(M)\)) of vector fields (resp. unit tangent bundles) on M. Contrary to the Sasaki metric on TM, where the two notions are equivalent to the harmonicity of the vector field and then to its parallelism, we prove that for large classes of g-natural metrics on TM, the two notions are not equivalent. Furthermore, we give examples of vector fields which are biharmonic as critical points of the bienergy functional restricted to \({\mathfrak {X}}(M)\), but are not biharmonic maps. We provide equally examples of proper biharmonic vector fields (resp. unit vector fields), i.e. those which are critical points of the bienergy functional restricted to the set \({\mathfrak {X}}(M)\) (resp. \({\mathfrak {X}}^1(M)\)) without being harmonic.

Area minimizing unit vector fields on antipodally punctured unit 2-sphere

We provide a lower value for the volume of a unit vector field tangent to an antipodally Euclidean sphere $\mathbb{S}^2$ depending on the length of an ellipse determined by the indexes of its singularities.

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 flat 3-manifold not equal to $\mathbb{E}^3$, 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 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.

Cyclic conformally flat hypersurfaces revisited

In this article we classify the conformally flat Euclidean hypersurfaces of dimension three with three distinct principal curvatures of $\mathbb{R}^4$, $\mathbb{S}^3\times \mathbb{R}$ and $\mathbb{H}^3\times \mathbb{R}$ with the property that the tangent component of the vector field $\partial/\partial t$ is a principal direction at any point. Here $\partial/\partial t$ stands for either a constant unit vector field in $\mathbb{R}^4$ or the unit vector field tangent to the factor $\mathbb{R}$ in the product spaces $\mathbb{S}^3\times \mathbb{R}$ and $\mathbb{H}^3\times \mathbb{R}$, respectively. Then we use this result to give a simple proof of an alternative classification of the cyclic conformally flat hypersurfaces of $\mathbb{R}^4$, that is, the conformally flat hypersurfaces of $\mathbb{R}^4$ with three distinct principal curvatures such that the curvature lines correspondent to one of its principal curvatures are extrinsic circles. We also characterize the cyclic conformally flat hypersurfaces of $\mathbb{R}^4$ as those conformally flat hypersurfaces of dimension three with three distinct principal curvatures for which there exists a conformal Killing vector field of $\mathbb{R}^4$ whose tangent component is an eigenvector field correspondent to one of its principal curvatures.

Fibrations of ℝ3 by oriented lines

A fibration of $\mathbb{R}^3$ by oriented lines is given by a unit vector field $V : \mathbb{R}^3 \to S^2$, for which all of the integral curves are oriented lines. A line fibration is called skew if no two fibers are parallel. Skew fibrations have been the focus of recent study, in part due to their close relationships with great circle fibrations of $S^3$ and with tight contact structures on $\mathbb{R}^3$. Both geometric and topological classifications of the space of skew fibrations have appeared; these classifications rely on certain rigid geometric properties exhibited by skew fibrations. Here we study these properties for line fibrations which are not necessarily skew, and we offer some partial answers to the question: in what sense do nonskew fibrations look and behave like skew fibrations? We develop and utilize a technique, called the parallel plane pushoff, for studying nonskew fibrations. In addition, we summarize the known relationship between line fibrations and contact structures, and we extend these results to give a complete correspondence. Finally, we develop a technique for generating nonskew fibrations and offer a number of examples.

Integral Formulas for a Foliation with a Unit Normal Vector Field

In this article, we prove integral formulas for a Riemannian manifold equipped with a foliation F and a unit vector field N orthogonal to F, and generalize known integral formulas (due to Brito-Langevin-Rosenberg and Andrzejewski-Walczak) for foliations of codimension one. Our integral formulas involve Newton transformations of the shape operator of F with respect to N and the curvature tensor of the induced connection on the distribution D=TF⊕span(N), and this decomposition of D can be regarded as a codimension-one foliation of a sub-Riemannian manifold. We apply our formulas to foliated (sub-)Riemannian manifolds with restrictions on the curvature and extrinsic geometry of the foliation.

Renormalized Energy Between Vortices in Some Ginzburg–Landau Models on 2-Dimensional Riemannian Manifolds

We study a variational Ginzburg–Landau type model depending on a small parameter \(\varepsilon >0\) for (tangent) vector fields on a 2-dimensional Riemannian manifold S. As \(\varepsilon \rightarrow 0\), these vector fields tend to have unit length so they generate singular points, called vortices, of a (non-zero) index if the genus \({\mathfrak {g}}\) of S is different than 1. Our first main result concerns the characterization of canonical harmonic unit vector fields with prescribed singular points and indices. The novelty of this classification involves flux integrals constrained to a particular vorticity-dependent lattice in the \(2{\mathfrak {g}}\)-dimensional space of harmonic 1-forms on S if \({\mathfrak {g}}\geqq 1\). Our second main result determines the interaction energy (called renormalized energy) between vortex points as a \(\Gamma \)-limit (at the second order) as \(\varepsilon \rightarrow 0\). The renormalized energy governing the optimal location of vortices depends on the Gauss curvature of S as well as on the quantized flux. The coupling between flux quantization constraints and vorticity, and its impact on the renormalized energy, are new phenomena in the theory of Ginzburg–Landau type models. We also extend this study to two other (extrinsic) models for embedded hypersurfaces \(S\subset {{\mathbb {R}}}^3\), in particular, to a physical model for non-tangent maps to S coming from micromagnetics.

A new construction on the energy of space curves in unit vector fields in Minkowski space E₂⁴

In this paper, we firstly introduce kinematics properties of a moving particle lying in Minkowski space E₂⁴. We assume that particles corresponds to different type of space curves such that they are characterized by Frenet frame equations. Guided by these, we present geometrical understanding of an energy and pseudo angle on the particle in each Frenet vector fields depending on the particle corresponds to a spacelike, timelike or lightlike curve in E₂⁴. Then we also determine the bending elastic energy functional for the same particle in E₂⁴ by assuming the particle has a bending feature of elastica. Finally, we prove that bending energy formula can be represented by the energy on the particle in each Frenet vector field.

A new quaternion valued frame of curves with an application

The aim of the paper is to obtain a new version of Serret-Frenet formulae for a quaternionic curve in R4 by using the method given by Bharathi and Nagaraj. Then, we define quaternionic helices in H named as quaternionic right and left X-helix with the help of given a unit vector field X. Since the quaternion product is not commutative, the authors ([4], [7]) have used by one-sided multiplication to find a space curve related to a given quaternionic curve in previous studies. Firstly, we obtain new expressions by using the right product and the left product for quaternions. Then, we generalized the construction of Serret-Frenet formulae of quaternionic curves. Finally, as an application, we obtain an example that supports the theory of this paper.

Existence of some classes of N(k)-quasi Einstein manifolds

The object of the present paper is to study some classes of N(k)-quasi Einstein manifolds. The existence of such manifolds are proved by giving non-trivial physical and geometrical examples. It is also proved that the characteristic vector field of the manifold is killing as well as parallel unit vector fields under certain curvaturerestrictions.

Detecting Object Surface Keypoints From a Single RGB Image via Deep Learning Network for 6-DoF Pose Estimation

Estimating the 6-DoF (Degree of Freedom) object pose from a single RGB image is one of the challenging tasks in the field of computer vision. Before the pose which is defined as the translation and rotation parameters can be derived by the traditional PnP algorithm, 2D image projections of a set of 3D object keypoints must be accurately detected. In this paper, we present techniques for defining 3D object surface keypoints and predicting their corresponding 2D counterparts via deep-learning network architectures. The main technique to designate 3D object keypoints is to employ quadratic fitting scheme for calculating the principal surface curvatures as the weights and then select from all surface points the ones mostly distributive with larger curvatures to describe the object shape as possible. However, the 2D projected keypoints are not directly regressed from the network, but encoded as the unit vector fields pointing to them, so that the voting scheme to recover back those 2D keypoints can be performed. Moreover, an effective loss function with the regularization term is adopted in training ResNet for predicting image projections of object keypoints by focusing on small-scale errors. Experimental results show that our proposed technique outperforms state-of-the-art approaches in both “2D projection” and “3D transformation” metrics.

Vacuum solutions in the Einstein–Aether Theory

The Einstein–Aether (EA) theory belongs to a class of modified gravity theories characterized by the introduction of a time-like unit vector field called aether. In this scenario, a preferred frame arises as a natural consequence of a broken Lorentz invariance. In the present work, we have obtained and analyzed some exact solutions allowed by this theory for two particular cases of a perfect fluid, both with Friedmann–Lemaître–Robertson–Walker symmetry: (i) a fluid with constant energy density (p = –ρ0) and (ii) a fluid with zero energy density (ρ0 = 0) corresponding to the vacuum solution with and without cosmological constant (Λ), respectively. Our solutions show that the EA and general relativity (GR) theories only differ in coupling constants. This difference is clearly shown because of the existence of singularities that are not in GR theory. This characteristic appears in the solutions with p = –ρ0 as well as with ρ0 = 0, where this last one depends only on the aether field. Furthermore, we consider the term of the EA theory in the Raychaudhuri equation and discuss the meaning of the strong energy condition in this scenario and found that this depends on the aether field. The solutions admit an expanding or contracting system. Bounce, singular, constant, and accelerated expansion solutions were also obtained, exhibiting the richness of the EA theory from the dynamic point of view of a collapsing system or of a cosmological model. The analysis of energy conditions, considering an effective fluid, shows that the term of the aether contributes significantly for the accelerated expansion of the system for the case in which the energy density is constant. On the other hand, for the vacuum case (ρ0 = 0), the energy conditions are all satisfied for the aether fluid.

Einstein Hypersurfaces of $$\mathbb {S}^n \times \mathbb {R}$$ and $$\mathbb {H}^n \times \mathbb {R}$$

In this paper, we classify the Einstein hypersurfaces of $$\mathbb {S}^n \times \mathbb {R}$$ and $$\mathbb {H}^n \times \mathbb {R}$$ . We use the characterization of the hypersurfaces of $$\mathbb {S}^n \times \mathbb {R}$$ and $$\mathbb {H}^n \times \mathbb {R}$$ whose tangent component of the unit vector field spanning the factor $$\mathbb {R}$$ is a principal direction and the theory of isoparametric hypersurfaces of space forms to show that Einstein hypersurfaces of $$\mathbb {S}^n \times \mathbb {R}$$ and $$\mathbb {H}^n \times \mathbb {R}$$ must have constant sectional curvature.

A Ginzburg–Landau Type Energy with Weight and with Convex Potential Near Zero

In this paper, we study the asymptotic behavior of minimizing solutions of a Ginzburg–Landau type functional with a positive weight and with convex potential near 0 and we estimate the energy in this case. We also generalize a lower bound for the energy of unit vector field given initially by Brezis–Merle–Rivière.

Interplay Between Loewner and Dirichlet Energies via Conformal Welding and Flow-Lines

The Loewner energy of a Jordan curve is the Dirichlet energy of its Loewner driving term. It is finite if and only if the curve is a Weil–Petersson quasicircle. In this paper, we describe cutting and welding operations on finite Dirichlet energy functions defined in the plane, allowing expression of the Loewner energy in terms of Dirichlet energy dissipation. We show that the Loewner energy of a unit vector field flow-line is equal to the Dirichlet energy of the harmonically extended winding. We also give an identity involving a complex-valued function of finite Dirichlet energy that expresses the welding and flow-line identities simultaneously. As applications, we prove that arclength isometric welding of two domains is sub-additive in the energy, and that the energy of equipotentials in a simply connected domain is monotone. Our main identities can be viewed as action functional analogs of both the welding and flow-line couplings of Schramm–Loewner evolution curves with the Gaussian free field.

A topological lower bound for the energy of a unit vector field on a closed Euclidean hypersurface

For a unit vector field on a closed immersed Euclidean hypersurface $M^{2n+1}$, $n\geq 1$, we exhibit a nontrivial lower bound for its energy which depends on the degree of the Gauss map of the immersion. When the hypersurface is the unit sphere $\mathbb

Mimetic Hořava gravity

We show that the scalar field of mimetic gravity could be used to construct diffeomorphism invariant models that reduce to Hořava gravity in the synchronous gauge. The gradient of the mimetic field provides a timelike unit vector field that allows to define a projection operator of four-dimensional tensors to three-dimensional spatial tensors. Conversely, it also enables us to write quantities invariant under space diffeomorphisms in fully covariant form without the need to introduce new propagating degrees of freedom.

Knots connected by wide ribbons

A ribbon is a smooth mapping (possibly self-intersecting) of an annulus [Formula: see text] in 3-space having constant width [Formula: see text]. Given a regular parametrization [Formula: see text], and a smooth unit vector field [Formula: see text] based along [Formula: see text], for a knot [Formula: see text], we may define a ribbon of width [Formula: see text] associated to [Formula: see text] and [Formula: see text] as the set of all points [Formula: see text], [Formula: see text]. For large [Formula: see text], ribbons, and their outer edge curves, may have self-intersections. In this paper, we analyze how the knot type of the outer ribbon edge [Formula: see text] relates to that of the original knot [Formula: see text]. Generically, as [Formula: see text], there is an eventual constant knot type. This eventual knot type is one of only finitely many possibilities which depend just on the vector field [Formula: see text]. The particular knot type within the finite set depends on the parametrized curves [Formula: see text], [Formula: see text], and their interactions. We demonstrate a way to control the curves and their parametrizations so that given two knot types [Formula: see text] and [Formula: see text], we can find a smooth ribbon of constant width connecting curves of these two knot types.