Normal Vector Field Research Articles

D-Brane effective Lagrangian in spacetimes with boundaries

AbstractIn this study, we explore the transformation of $$D_p$$ D p -branes to $$D_{p-1}$$ D p - 1 -branes under T-duality when the D-brane is embedded in a spacetime with a boundary. Our goal is to derive the higher-derivative corrections to the Dirac–Born–Infeld (DBI) Lagrangian for both the bulk and boundary terms. For the bulk terms, we calculate the $$\alpha '$$ α ′ corrections for the massless open string fields, up to the 8th order in the dimensionless Maxwell field strength. We demonstrate that the bulk Lagrangian can satisfy the T-duality constraint without residual total derivative terms in the base space. This determines the most general independent couplings of the massless open string fields in the bulk Lagrangian, encompassing 145 coupling constants, up to three parameters. Two of these parameters are physical and are determined by disk-level S-matrix elements, while the third is unphysical and can be eliminated by field redefinitions and integration by parts. The final result for the bulk Lagrangian consists of 49 couplings. For the boundary terms, the application of T-duality symmetry to massless open and closed string fields enables us to extend the DBI Lagrangian to include the boundary unit normal vector field $$n_\mu $$ n μ and its first derivative.

A space level light bulb theorem in all dimensions

Given a d -dimensional manifold M and a knotted sphere s\colon \mathbb{S}^{k-1}\hookrightarrow \partial M with 1\leq k\leq d , for which there exists a framed dual sphere G\colon \mathbb{S}^{d-k}\hookrightarrow \partial M , we show that the space of neat embeddings \mathbb{D}^{k} \hookrightarrow M with boundary s can be delooped by the space of neatly embedded (k-1) -disks, with a normal vector field, in the d -manifold obtained from M by attaching a handle to G . This increase in codimension significantly simplifies the homotopy type of such embedding spaces, and is of interest also in low-dimensional topology. In particular, we apply the work of Dax to describe the first interesting homotopy group of these embedding spaces, in degree d-2k . In a separate paper we use this to give a complete isotopy classification of 2-disks in a 4-manifold with such a boundary dual.

On tiling spherical triangles into quadratic subpatches

Various interpolation and approximation methods arising in several practical applications in geometric modeling deal, at a particular step, with the problem of computing suitable rational patches (of low degree) on the unit sphere. Therefore, we are concerned with the construction of a system of spherical triangular patches with prescribed vertices that globally meet along common boundaries. In particular, we investigate various possibilities for tiling a given spherical triangular patch into quadratically parametrizable subpatches. We revisit the condition that the existence of a quadratic parameterization of a spherical triangle is equivalent to the sum of the interior angles of the triangle being π, and then circumvent this limitation by studying alternative scenarios and present constructions of spherical macro-elements of the lowest possible degree. Applications of our method include algorithms relying on the construction of (interpolation) surfaces from prescribed rational normal vector fields.

In situ build surface topography determination in electron beam powder bed fusion

Electron optical imaging is the most promising process monitoring method in electron beam powder bed fusion. State of the art in modern machines is the installation of a single detector in the top center of the build chamber. Exemplary applications are the reconstruction of digital twins of manufactured parts to compare their dimensional accuracy or analysing the top surface of each layer to identify surface features like pores or material transport. Multi-detector systems are currently under research and have shown great potential in reconstructing the surface topography in situ. A recently developed ray tracing model, describing the image formation process, allows to formulate design guide lines for multi-detector systems and provides a method for the computation of the normal vector field of the build surface. This work utilizes the recent progress and presents a newly developed four-detector system and an updated computation chain, which enable build surface topography reconstruction in situ in every layer of a build process. The computation chain contains a normal integration algorithm, which employs Tikhonov regularization to cope with measurement irregularities. The integration method is validated with ex situ measured as-built surfaces. Additionally, first applications are demonstrated and connections to process parameter changes illustrated.

A New Representation for Slant Curves in Sasakian 3-Manifolds

In this paper, we define a new kind of curve called $N$-slant curve whose principal normal vector field makes a constant angle with the Reeb vector field $\xi$ in Sasakian $3$-manifolds. Then, we give some characterizations of $N$-slant curves in Sasakian $3$-manifolds and we obtain some properties of the curves in $\mathbb{R}^{3}(-3)$. Moreover, we investigate the conditions of $C$-parallel and $C$-proper mean curvature vector fields along $N$-slant curves in Sasakian $3$-manifolds. Finally, we study $N$-slant curves of type $AW(k)$ where k=1,2 or 3.

Point cloud denoising using a generalized error metric

Effective removal of noises from raw point clouds while preserving geometric features is the key challenge for point cloud denoising. To address this problem, we propose a novel method that jointly optimizes the point positions and normals. To preserve geometric features, our formulation uses a generalized robust error metric to enforce piecewise smoothness of the normal vector field as well as consistency between point positions and normals. By varying the parameter of the error metric, we gradually increase its non-convexity to guide the optimization towards a desirable solution. By combining alternating minimization with a majorization-minimization strategy, we develop a numerical solver for the optimization which guarantees convergence. The effectiveness of our method is demonstrated by extensive comparisons with previous works.

The Polya f-curvature of plane curves

We introduce and study a new curvature function for plane curves inspired by the weighted mean curvature of M. Gromov. We call it Pólya, being the difference between the usual curvature and the inner product of the normal vector field with the Pólya vector field of a given planar function f. We computed it for several examples, since the general problem of vanishing or constant values of this new curvature involves the general expression of f. Mathematics Subject Classification (2010): 53A04, 53A45, 53A55. Received 03 May 2023; Accepted 17 January 2024

Codimension two spacelike submanifolds in Lorentzian manifolds and conformal structures

Starting from a Riemannian conformal structure on a manifold M, we provide a method to construct a family of Lorentzian manifolds. The construction relies on the choice of a metric in the conformal class and a smooth 1-parameter family of self-adjoint tensor fields. Then, every metric in the conformal class corresponds to the induced metric on M seen as a codimension two spacelike submanifold into these Lorentzian manifolds. Under suitable choices of the 1-parameter family of tensor fields, there exists a lightlike normal vector field along such spacelike submanifolds whose Weingarten endomorphism provide a Möbius structure on the Riemannian conformal structure. Conversely, every Möbius structure on a Riemannian conformal structure arises in this way. Flat Möbius structures are characterized in terms of the extrinsic geometry of the corresponding spacelike surfaces.

A normal line congruence and minimal ruled Lagrangian submanifolds in [formula omitted

We characterize Lagrangian submanifolds in complex projective space for which each parallel submanifold along normal geodesics with respect to a unit normal vector field is Lagrangian, by using a normal line congruence of the Lagrangian submanifold to complex 2-plane Grassmannian and quaternionic Kähler structure. As a special case, we can construct minimal ruled Lagrangian submanifolds in complex projective space from an austere hypersurface in sphere with non-vanishing Gauss-Kronecker curvature.

Phase matching of electromagnetic wave on moving interface

Due to the boundary conditions of electromagnetic fields and phase matching of electro magnetic waves on interface being the basis to drive the Snell’s laws and Fresnel’s laws, they are also crucial for the analysis of electromagnetic wave propagation in a moving medium. There are mainly two methods to derive the boundary conditions of electromagnetic fields on moving interface. One of them is to use the kinematic integral form, yet this method is based on the classical time-space, and the other is based on the relativistic transformation, the boundary conditions are derived from the scaling effect with geometric method, or from the principle of relativity directly. However, the first one has a form the same as the form obtained by using the kinematic integral form, while the second one obtains a different form. At the same time, the phase matching of electromagnetic wave on moving interface is only discussed by Galileo transformation, however this is unreasonable, because of the relativistic effect cannot be ignored here. Therefore, it is necessary to reexamine the boundary conditions of electromagnetic fields and phase matching of electromagnetic wave on moving interface. Herein, firstly, the relativistic transformation formula of the unit normal vector of moving surface is derived from the surface equation and principle of relativity. Secondly, the boundary conditions of electromagnetic fields on moving interface are given based on the relativistic transformation formula and the non-relativistic transformation formula of the unit normal vector and electromagnetic fields, which show that the boundary conditions of electromagnetic fields on moving interface under the relativistic case and the non-relativistic case have the same form. This is not accidental but definite, because the change of flux of electromagnetic fields, like the change of magnetic flux, from the induction of electromagnetic filed is the same as that from the variation of surface element. Thirdly, the phase matching of electromagnetic wave on moving interface is given based on the relativistic transformation formula of the unit normal vector and the phase matching of electromagnetic wave on resting interface. In the problem of light incident on a homogeneous medium moving at a constant velocity in vacuum or air, using the phase matching of electromagnetic wave on moving interface, the same results can be easily obtained through other methods. The discussion in this study belongs to classical electrodynamics with no quantum effects considered, but the results will provide some conveniences for theoretically analyzing electromagnetic communication, remote sensing and telemetering.

Construction of new Lie group and its geometric properties

<abstract><p>In this paper, we constructed a novel Lie group by using oblate spheroidal coordinates. First, we took the metric tensor of oblate spheroidal coordinates, then found its Killing vectors by using the Killing equation. After solving a system of partial differential equations, we obtained the Killing vectors. With the help of these Killing vectors, we first constructed finite Lie algebra and then proved that Killing vectors form a Lie group. Also, we described the geometric properties in which this Lie group forms a regular surface, defined the differential map and differential of normal vector field, and found the gaussian and mean curvatures.</p></abstract>

Ruled real hypersurfaces in the complex hyperbolic quadric

Abstract In this article, we introduce a new family of real hypersurfaces in the complex hyperbolic quadric Q n ∗ = S O 2 , n o ∕ S O 2 S O n {{Q}^{n}}^{\ast }=S{O}_{2,n}^{o}/S{O}_{2}S{O}_{n} , namely, the ruled real hypersurfaces foliated by complex hypersurfaces. Berndt described an example of such a real hypersurface in Q n ∗ {{Q}^{n}}^{\ast } as a homogeneous real hypersurface generated by a A {\mathfrak{A}} -principal horocycle in a real form R H n {\mathbb{R}}{H}^{n} . So, in this article, we compute a detailed expression of the shape operator for ruled real hypersurfaces in Q n ∗ {{Q}^{n}}^{\ast } and investigate their characterizations in terms of the shape operator and the integrable distribution C = { X ∈ T M ∣ X ⊥ ξ } {\mathcal{C}}=\left\{X\in TM| X\perp \xi \right\} . Then, by using these observations, we give two kinds of classifications of real hypersurfaces in Q n ∗ {{Q}^{n}}^{\ast } satisfying η \eta -parallelism under either η \eta -commutativity of the shape operator or integrability of the distribution C {\mathcal{C}} . Moreover, we prove that the unit normal vector field of a real hypersurface with η \eta -parallel shape operator in Q n ∗ {{Q}^{n}}^{\ast } is A {\mathfrak{A}} -principal. On the other hand, it is known that all contact real hypersurfaces in Q n ∗ {{Q}^{n}}^{\ast } have a A {\mathfrak{A}} -principal normal vector field. Motivated by these results, we give a characterization of contact real hypersurfaces in Q n ∗ {{Q}^{n}}^{\ast } in terms of η \eta -parallel shape operator.

Global rigorous coupled wave analysis for design of multilayer metasurface absorbers.

In this work, a novel Global NV-ETM RCWA method is proposed to accelerate the optimization of the periodic stepped radar absorbing structure. This method is based on the rigorous coupled-wave analysis (RCWA) utilizing the normal vector field (NV) and enhanced transmittance matrix (ETM) approach. The NV field dramatically improves the convergence rate for both dielectric and magnetic metasurfaces. The Global NV-ETM RCWA algorithm is developed to further accelerate the complete search calculations. Using the proposed method, the periodic stepped radar absorbing structures are efficiently optimized to realize the entire band absorption in 2-18 GHz. The optimization results demonstrate the Global NV-ETM RCWA method significantly increase the computational efficiency, with a 38-fold improvement over direct NV-ETM RCWA calculations when the truncation order N=3. This method provides a powerful tool for designing metasurface absorbers with various desired functionalities.

The lifts of surfaces in neutral 4-manifolds into the 2-Grassmann bundles

A twistor lift of a space-like or time-like surface in a neutral hyperKähler 4-manifold with zero mean curvature vector is given by a (para)holomorphic function, which yields (para)holomorphicity of the Gauss maps of space-like or time-like surfaces in E24 with zero mean curvature vector. For a space-like or time-like surface in an oriented neutral 4-manifold with zero mean curvature vector such that both twistor lifts belong to the kernel of the curvature tensor, its (para)complex quartic differential is holomorphic. If both twistor lifts of a time-like surface with zero mean curvature vector have light-like or zero covariant derivatives, then either the shape operator with respect to a light-like normal vector field vanishes or all the shape operators of the surface are light-like or zero. Examples with the former (resp. latter) property are given by the conformal Gauss maps of time-like surfaces of Willmore type with zero paraholomorphic quartic differential (resp. time-like surfaces in 4-dimensional neutral space forms based on the Gauss-Codazzi-Ricci equations).

Entire solutions to 4 dimensional Ginzburg–Landau equations and codimension 2 minimal submanifolds

We consider the magnetic Ginzburg–Landau equations in R4{−ε2(∇−iA)2u=12(1−|u|2)u,ε2d⁎dA=〈(∇−iA)u,iu〉, formally corresponding to the Euler–Lagrange equations for the energy functionalE(u,A)=12∫R4|(∇−iA)u|2+ε2|dA|2+14ε2(1−|u|2)2. Here u:R4→C, A:R4→R4 and d denotes the exterior derivative acting on the one-form dual to A. Given a minimal surface M2 in R3 with finite total curvature and non-degenerate, we construct a solution (uε,Aε) which has a zero set consisting of a smooth surface close to M×{0}⊂R4. Away from the latter surface we have |uε|→1 anduε(x)→z|z|,Aε(x)→1|z|2(−z2ν(y)+z1e4),x=y+z1ν(y)+z2e4 for all sufficiently small z≠0. Here y∈M and ν(y) is a unit normal vector field to M in R3.

A Meshfree Collocation Scheme for Surface Differential Operators on Point Clouds

We present a meshfree collocation scheme to discretize intrinsic surface differential operators over scalar fields on smooth curved surfaces with given normal vectors and a non-intersecting tubular neighborhood. The method is based on discretization-corrected particle strength exchange (DC-PSE), which generalizes finite difference methods to meshfree point clouds. The proposed Surface DC-PSE method is derived from an embedding theorem, but we analytically reduce the operator kernels along surface normals to obtain a purely intrinsic computational scheme over surface point clouds. We benchmark Surface DC-PSE by discretizing the Laplace–Beltrami operator on a circle and a sphere, and we present convergence results for both explicit and implicit solvers. We then showcase the algorithm on the problem of computing Gauss and mean curvature of an ellipsoid and of the Stanford Bunny by approximating the intrinsic divergence of the normal vector field. Finally, we compare Surface DC-PSE with surface finite elements (SFEM) and diffuse-interface finite elements (DI FEM) in a validation case.

Cyclic semiparallel Ricci tensor for harmonic curvature in the complex quadric

AbstractWe introduce the notion of harmonic curvature for real hypersurfaces in the complex quadric . First, we prove that the unit normal vector field N on real hypersurfaces in with harmonic curvature is singular, that is, ‐principal or ‐isotropic. Next by using a new notion of cyclic semiparallel Ricci tensor, we give a new result on real hypersurfaces with harmonic curvature and constant mean curvature in .

Codimension 2 submanifolds of paracosymplectic manifolds with normal Reeb vector field

This study is devoted to a submanifold M of codimension 2 of an almost paracontact metric manifold M, for which the Reeb vector field of the ambient manifold is normal. Some sufficient conditions for the existence of M are given. When M is paracosymplectic, then some necessary and sufficient conditions are established for M to fall in one of the following classes of almost paracontact metric manifolds according to the classification given by S. Zamkovoy and G. Nakova: normal, paracontact metric, para-Sasakian, K-paracontact, quasi-para-Sasakian, respectively. When in addition, M is para-Sasakian and M is paracosymplectic, some characterization results are obtained for M to be totally umbilical, as well as a nonexistence result for M to be totally geodesic is provided. The case when M is of a constant sectional curvature is analysed and an example is constructed.

*-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.

$ G^2/C^1 $ Hermite interpolation of offset curves of parametric regular curves

<abstract><p>In this paper we presented a method of $ G^2 $ Hermite interpolation of offset curves of regular plane curves based on approximating the normal vector fields. We showed that our approximant is also $ C^1 $ Hermite interpolation of the offset curve. Our method is capable of achieving circular precision. Another advantage of our method is that if the input curve is a polynomial curve, then our method also yields a polynomial curve. Our approximation method was applied to numerical examples and its numerical results were compared to previous offset approximation methods. It was observed that our method is almost optimal with respect to the number of control points of the approximation curves for the same tolerance.</p></abstract>