Radial Vector Field Research Articles

Conformal trajectories in three-dimensional space forms

We introduce the notion of conformal trajectories in three-dimensional Riemannian manifolds [Formula: see text]. Given a conformal vector field [Formula: see text], a conformal trajectory of [Formula: see text] is a regular curve [Formula: see text] in [Formula: see text] satisfying [Formula: see text], for some fixed nonzero constant [Formula: see text]. In this paper, we study the conformal trajectories in the space forms [Formula: see text], [Formula: see text] and [Formula: see text]. For (non-Killing) conformal vector fields in [Formula: see text] (respectively, in [Formula: see text]), we prove that conformal trajectories have constant curvature and its torsion is a linear combination of trigonometric (respectively, hyperbolic) functions on the arc-length parameter. In the case of Euclidean space [Formula: see text], we obtain the same result for the radial vector field and characterizing all conformal trajectories.

Coupled vector Gauss-Bonnet theories and hairy black holes

We study vector-tensor theories in which a 4-dimensional vector field Aμ is coupled to a vector quantity Jμ, which is expressed in terms of Aμ and a metric tensor gμν. The divergence of Jμ is equivalent to a Gauss-Bonnet (GB) term. We show that an interacting Lagrangian of the form f(X)AμJμ, where f is an arbitrary function of X=−(1/2)AμAμ, belongs to a scheme of beyond generalized Proca theories. For f(X)=α=constant, this interacting Lagrangian reduces to a particular class of generalized Proca theories. We apply the latter coupling to a static and spherically symmetric vacuum configuration by incorporating the Einstein-Hilbert term, Maxwell scalar, and vector mass term ηX (η is a constant). Under an expansion of the small coupling constant α with η≠0, we derive hairy black hole solutions endowed with nonvanishing temporal and radial vector field profiles. The asymptotic properties of solutions around the horizon and at spatial infinity are different from those of hairy black holes present in scalar-GB theories. We also show that black hole solutions without the vector mass term, i.e., η=0, are prone to ghost instability of odd-parity perturbations.

Short Proofs of Explicit Formulas to Boundary Value Problems for Polyharmonic Equations Satisfying Lopatinskii Conditions

This paper deals with Lopatinskii type boundary value problem (bvp) for the (poly) harmonic differential operators. In the case of Robin bvp for the Laplace equation in the ball B1 a Green function is constructed in the cases c>0, c∉−N, where c is the coefficient in front of u in the boundary condition ∂u∂n+cu=f. To do this a definite integral must be computed. The latter is possible in quadratures (elementary functions) in several special cases. The simple proof of the construction of the Green function is based on some solutions of the radial vector field equation Λu+cu=f. Elliptic boundary value problems for Δmu=0 in B1 are considered and solved in Theorem 2. The paper is illustrated by many examples of bvp for Δu=0, Δ2u=0 and Δ3u=0 in B1 as well as some additional results from the theory of spherical functions are proposed.

Harmonic radial vector fields on harmonic spaces

We characterize harmonic spaces in terms of the dimensions of various spaces of radial eigen-spaces of the Laplacian Δ0 on functions and the Laplacian Δ1 on 1-forms. We examine the nature of the singularity as the geodesic distance r tends to zero of radial eigen-functions and 1-forms. Via duality, our results give rise to corresponding results for radial vector fields.

An intrinsic hyperboloid approach for Einstein Klein–Gordon equations

Klainerman introduced in [7] the hyperboloidal method to prove global existence results for nonlinear Klein–Gordon equations by using commuting vector fields. In this paper, we extend the hyperboloidal method from Minkowski space to Lorentzian spacetimes. This approach is developed in [15] for proving, under the maximal foliation gauge, the global nonlinear stability of Minkowski space for Einstein equations with massive scalar fields, which states that, the sufficiently small data in a compact domain, surrounded by a Schwarzschild metric, leads to a unique, globally hyperbolic, smooth and geodesically complete solution to the Einstein Klein–Gordon system. In this paper, we set up the geometric framework of the intrinsic hyperboloid approach in the curved spacetime. By performing a thorough geometric comparison between the radial normal vector field induced by the intrinsic hyperboloids and the canonical $\partial_r$, we manage to control the hyperboloids when they are close to their asymptote, which is a light cone in the Schwarzschild zone. By using such geometric information, we not only obtain the crucial boundary information for running the energy method in [15], but also prove that the intrinsic geometric quantities including the Hawking mass all converge to their Schwarzschild values when approaching the asymptote.

A Geometric Approach to the Concept of Extensivity in Thermodynamics

This paper presents a rigorous treatment of the concept of extensivity in equilibrium thermodynamics from a geometric point of view. This is achieved by endowing the manifold of equilibrium states of a system with a smooth atlas that is compatible with the pseudogroup of transformations on a vector space that preserve the radial vector field. The resulting geometric structure allows for accurate definitions of extensive differential forms and scaling, and the well-known relationship between both is reproduced. This structure is represented by a global vector field that is locally written as a radial one. The submanifolds that are transversal to it are embedded, and locally defined by extensive functions.

径向矢量光场在双延迟器作用下的偏振演化

径向矢量光场在双延迟器作用下的偏振演化

Dirichlet principal eigenvalue comparison theorems in geometry with torsion

We describe min-max formulas for the principal eigenvalue of a V-drift Laplacian defined by a vector field V on a geodesic ball of a Riemannian manifold N. Then we derive comparison results for the principal eigenvalue with the one of a spherically symmetric model space endowed with a radial vector field, under pointwise comparison of the corresponding radial sectional and Ricci curvatures, and of the radial component of the vector fields. These results generalize the known case V=0.

Multi-Center Vector Field Methods for Wave Equations

We develop the method of vector-fields to further study Dispersive Wave Equations. Radial vector fields are used to get a-priori estimates such as the Morawetz estimate on solutions of Dispersive Wave Equations. A key to such estimates is the repulsiveness or nontrapping conditions on the flow corresponding to the wave equation. Thus this method is limited to potential perturbations which are repulsive, that is the radial derivative pointing away from the origin. In this work, we generalize this method to include potentials which are repulsive relative to a line in space (in three or higher dimensions), among other cases. This method is based on constructing multi-centered vector fields as multipliers, cancellation lemmas and energy localization.

Vector fields whose linearisation is Hurwitz almost everywhere

A real matrix is Hurwitz if its eigenvalues have negative real parts. The following generalisation of the Bidimensional Global Asymptotic Stability Problem (BGAS) is provided. Let X : R 2 → R 2 X:\mathbb {R}^2\to \mathbb {R}^2 be a C 1 C^1 vector field whose Jacobian matrix D X ( p ) DX(p) is Hurwitz for Lebesgue almost all p ∈ R 2 p\in \mathbb {R}^2 . Then the singularity set of X X is either an empty set, a one–point set or a non-discrete set. Moreover, if X X has a hyperbolic singularity, then X X is topologically equivalent to the radial vector field ( x , y ) ↦ ( − x , − y ) (x,y)\mapsto (-x,-y) . This generalises BGAS to the case in which the vector field is not necessarily a local diffeomorphism.

Poincaré-Hopf theorems on singular spaces

In 1991, Schwartz proved a Poincaré–Hopf theorem for stratified radial vector fields on Whitney stratified analytic varieties with boundary. In this paper, we extend her theory and results to a wide class of stratified sets with boundary, called radial manifold complexes, which include all Thom–Mather stratified sets, but also arbitrary closure orderable finite partitions of subanalytic sets into subanalytic submanifolds. In the process, we give a short proof of a generalization of the Poincaré–Hopf theorem for radial vector fields of Schwartz (our Theorem 3.4). We then study more general stratified vector fields that need not be radial and may further be tangent to the boundary of the manifold complex, and prove the corresponding Poincaré–Hopf theorems. In particular, a notion is given of the virtual index of a vector field on a radial manifold complex and the Euler characteristic of the complex is shown to be the sum of the virtual indices.

Shape and period of limit cycles bifurcating from a class of Hamiltonian period annulus

In this work we are concerned with the problem of shape and period of isolated periodic solutions of perturbed analytic radial Hamiltonian vector fields in the plane. Françoise developed a method to obtain the first non vanishing Poincaré–Pontryagin–Melnikov function. We generalize this technique and we apply it to know, up to any order, the shape of the limit cycles bifurcating from the period annulus of the class of radial Hamiltonians. We write any solution, in polar coordinates, as a power series expansion in terms of the small parameter. This expansion is also used to give the period of the bifurcated periodic solutions. We present the concrete expression of the solutions up to third order of perturbation of Hamiltonians of the form H=H(r). Necessary and sufficient conditions that show if a solution is simple or double are also presented.

On Some Finsler Metrics of Non-positive Constant Flag Curvature

(α, β)-norms on \({\mathbb{R}^N}\) induce Minkowski metrics, and the construction of related homothetic vector fields gives a family of new Finsler metrics of non-positive constant flag curvature for each non-trivial (α, β)-norm. The dimension of this family is at least \({\tfrac{1}{2}(N^2 - N + 4)}\) . In particular, we generalize the Funk metric on the unit ball via navigation representation of the standard Euclidean norm and the radial vector field. Finally, we describe the geodesics of these new Finsler metrics with constant flag curvature.

Trapping Horizons in Sultana—Dyer Space-Time

The Sultana—Dyer space-time is suggested as a model describing a black hole embedded in an expanding universe. Recently, its global structure is analyzed and the trapping horizons are shown. In the paper, by directly calculating the expansions of the radial null vector fields normal to the space-like two-spheres foliating the trapping horizons, we find that the trapping horizon outside the event horizon in the Sultana—Dyer space-time is a past trapping horizon. Further, we find that the past trapping horizon is an outer, instantaneously degenerate or inner trapping horizon accordingly when the radial coordinate is less than, equal to or greater than some value.

Energy and topology of singular unit vector fields on S3

The energy of unit vector fields on odd-dimensional spheres is a functional that has a minimum in dimension 3 and an infimum in higher dimensions. Vector fields with isolated singularities arise naturally in the study of this functional. We consider the class of fields in S3 having two antipodal singularities. We prove a lower bound, attained for the radial vector field, for the energy of this class of fields in terms of the indices of the singularities. A similar inequality is not to be expected in other dimensions.

Normal forms of vector fields on Poisson manifolds

We study formal and analytic normal forms of radial and Hamiltonian vector fields on Poisson manifolds near a singular point.

Determining the Geometry of Boundaries of Objects from Medial Data

We consider a region ? in R2 or R3 with generic smooth boundary B and Blum medial axis M, on which is defined a multivalued vector field U from points x on M to the points of tangency of the sphere at x with B. We introduce a shape operator Srad and an edge shape operator SE which measure how U bends along M. These are not traditional differential shape operators, nonetheless we derive all local differential invariants of B from these operators. This allows us to define from (M, U) a geometric medial on M which corresponds, via a from M to B, to the differential properties of B. The medial map also includes a description of the geometry of B. This is defined using the relative critical of the radius function r on M. This set consists of a network of curves on M which describe where B is thickest and thinnest. It is computed using the covariant derivative of the tangential component of the unit radial vector field. We further determine how these invariants are related to the differential invariants of M and how these invariants change under deforming diffeomorphisms of M.

Smoothness and geometry of boundaries associated to skeletal structures, II: Geometry in the Blum case

A skeletal structure (M, U) in . In this paper, in the partial Blum case we derive formulas for the differential geometric shape operator of the boundary (and hence all curvature invariants) in terms of the shape operators on the medial axis. We further derive the effects of a diffeomorphism of the skeletal structure on the radial and edge shape operators using a distortion operator which is computed from the second derivative of the diffeomorphism evaluated on the unit radial vector field. This allows one to compute the geometry of the boundary associated to a deformed skeletal structure purely in terms of operators defined on the original skeletal set.

Harmonic and minimal unit vector fields on Riemannian symmetric spaces

We present new examples of harmonic and minimal unit vector fields on Riemannian symmetric spaces. These examples are constructed from cohomogeneity one actions with a reflective singular orbit. The radial unit vector field associated to such a reflective submanifold is harmonic and minimal.

The Infimum of the Energy of Unit Vector Fields on Odd-Dimensional Spheres

We construct a one-parameter family of unit smooth vector fieldsglobally defined on the sphere \(\mathbb{S}\)2k+1 for k ≥ 2, with energyconverging to the energy of the unit radial vector field, which isdefined on the complementary of two antipodal points. So we prove thatthe infimum of the energy of globally defined unit smooth vector fieldsis $$\left( {\frac{{2k + 1}}{2} + \frac{k}{{2k - 1}}} \right){\text{ vol (}}\mathbb{S}^{2k + 1} ).$$