Normal Vector Field Research Articles

Derivatives of normal Jacobi operator on real hypersurfaces in the complex quadric

In \cite{S 2017}, Suh gave a non-existence theorem for Hopf real hypersurfaces in the complex quadric with parallel normal Jacobi operator. Motivated by this result, in this paper, we introduce some generalized conditions named $\mathcal C$-parallel or Reeb parallel normal Jacobi operators. By using such weaker parallelisms of normal Jacobi operator, first we can assert a non-existence theorem of Hopf real hypersurfaces with $\mathcal C$-parallel normal Jacobi operator in the complex quadric $Q^{m}$, $m \geq 3$. Next, we prove that a Hopf real hypersurface has Reeb parallel normal Jacobi operator if and only if it has an $\mathfrak A$-isotropic singular normal vector field.

Rigidity of parabolic and stable spacelike hypersurfaces in a generalized Robertson–Walker spacetime

In this paper, we deal with a parabolic stable spacelike hypersurface immersed in a generalized Robertson–Walker spacetime. We consider the support type function as the product between the warping function and the hyperbolic cosine of the hyperbolic angle of the normal vector field with the timelike parallel direction $$\partial _t$$ . We provide some conditions under which the support type function must be constant or, particularly, when the hypersurface must be a slice, yielding a Calabi–Bernstein type result.

Real Hypersurfaces in $$Q^m$$ with Commuting Structure Jacobi Operator

In this paper, we study real hypersurfaces in the complex quadric space $$Q^m$$ whose structure Jacobi operator commutes with their structure tensor field. When the normal vector field is $$\mathfrak {A}$$ -principal we show that the Reeb curvature $$\alpha $$ is non-vanishing and determine principal curvatures of the hypersurface. In the case of $$\mathfrak {A}$$ -isotropic normal vector field, we prove that the hypersurface is Hopf if it has vanishing Reeb curvature or commuting shape operator. We also consider Reeb flat hypersurfaces, namely when the Reeb curvature is zero. We see that this family of hypersurfaces is non-empty and among other results we prove that if the Ricci tensor of a Reeb flat Hopf hypersurfaces is Killing, then the Ricci tensor is parallel.

Contact CR Submanifolds of maximal Contact CR dimension of Sasakian Space Form

In this paper, we investigate contact CR submanifolds of contact CR dimension in Sasakian space form and introduce the general structure of these submanifolds and then studying structures of this submanifols with the condition  h(FX,Y)+h(X,FY)=g(FX,Y)zeta, for the normal vector field zeta, which is nonzero, and we classify these submanifolds../files/site1/files/61/0Abstract.pdf

Total variation of the normal vector field as shape prior

An analogue of the total variation prior for the normal vector field along the boundary of smooth shapes in 3D is introduced. The analysis of the total variation of the normal vector field is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere manifold. It is shown that spheres are stationary points when the total variation of the normal is minimized under an area constraint. Shape calculus is used to characterize the relevant derivatives. Since the total variation functional is non-differentiable whenever the boundary contains flat regions, an extension of the split Bregman method to manifold valued functions is proposed.

Discrete total variation of the normal vector field as shape prior with applications in geometric inverse problems

An analogue of the total variation prior for the normal vector field along the boundary of piecewise flat shapes in 3D is introduced. A major class of examples are triangulated surfaces as they occur for instance in finite element computations. The analysis of the functional is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere manifold. It is found to agree with the discrete total mean curvature known in discrete differential geometry. A split Bregman iteration is proposed for the solution of discretized shape optimization problems, in which the total variation of the normal appears as a regularizer. Unlike most other priors, such as surface area, the new functional allows for piecewise flat shapes. As two applications, a mesh denoising and a geometric inverse problem of inclusion detection type involving a partial differential equation are considered. Numerical experiments confirm that polyhedral shapes can be identified quite accurately.

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.

Differential geometry of immersed surfaces in three-dimensional normed spaces

In this paper we study curvature types of immersed surfaces in three-dimensional (normed or) Minkowski spaces. By endowing the surface with a normal vector field, which is a transversal vector field given by the ambient Birkhoff orthogonality, we get an analogue of the Gauss map. Then we can define concepts of principal, Gaussian, and mean curvatures in terms of the eigenvalues of the differential of this map. Considering planar sections containing the normal field, we also define normal curvatures at each point of the surface, and with respect to each tangent direction. We investigate the relations between these curvature types. Further on we prove that, under an additional hypothesis, a compact, connected surface without boundary whose Minkowski Gaussian curvature is constant must be a Minkowski sphere. Since existing literature on the subject of our paper is widely scattered, in the introductory part also a survey of related results is given.

Construction of the spacelike constant angle surface family in Minkowski 3-space

A spacelike constant angle surface in $E_1^3$ (Minkowski 3-space) is defined by its unit normal vector field forms a constant angle with some fixed vector direction. In this paper, we use the Frenet frame to express the surface and construct a family of spacelike constant angle surfaces which possess the given curve as isoparametric curve. The normal vector of each surface forms a constant angle with a fixed vector lying in the spacelike cone and in the timelike cone respectively. Finally, we give some representative examples.

Discrete Normal Vector Field Approximation via Time Scale Calculus

Abstract The theory of time scales calculus have long been a subject to many researchers from different disciplines. Beside the unification and the extension aspects of the theory, it emerge as a powerful tool for mimetic discretization process. In this study, we present a framework to find normal vector fields of discrete point sets in ℝ3 by using symmetric differential on time scales. A surface parameterized by the tensor product of two time scales can be analogously expressed as the vertex set of non-regular rectangular grids. If the time scales are dense, then the discrete grid structure vanishes. If the time scales are isolated, then the further geometric analysis can be executed by using symmetric dynamic differential. Moreover, we present an algorithmic procedure to determine the symmetric dynamic differential structure on the neighborhood of points in surfaces. Our results indicate that the method we present has good approximation to unit normal vector fields of parameterized surfaces rather than the Delaunay triangulation for some points.

Construction of Normal Vector Field Using the Partial Differential Equations

Construction of Normal Vector Field Using the Partial Differential Equations

Geometry Processing Problems Using the Total Variation of the Normal Vector Field

AbstractThis paper presents a novel approach to the formulation and solution of discrete geometry processing problems including mesh denoising. The main quantity of interest is the piecewise constant unit normal vector field, which we consider on piecewise flat, triangulated surfaces. In a similar fashion as one does for images defined on ‘flat’ domains, our goal is to remove noise while preserving shape features such as sharp edges. To this end, we model a denoising problem via a quadratic vertex tracking term and a regularizer based on the total variation of the normal vector field. Since the latter has values on the unit sphere, its total variation reduces to a finite sum of the geodesic distances of neighboring normals. We solve the model numerically by applying split‐Bregmann iterations and provide results for the ‘fandisk’ benchmark.

The slant helices according to N-Bishop frame of the spacelike curve with spacelike principal normal in Minkowski 3-space

If the principal normal vector field of a curve makes a constant angle with constant direction, this curve is called as slant helix. In this paper, a slant helix is defined according to N-Bishop frame of the spacelike curve with a spacelike principal normal. Some characterizations of the slant helices are obtained according to spacelike curve N-Bishop frame with a spacelike principal normal, benefiting from the definition of the slant helices.

Parametric Equations for Space Curves Whose Spherical Images Are Slant Helices

The curve whose tangent and binormal indicatrices are slant helices is called a slant-slant helix.
 
 In this paper, we give a new characterization of a slant-slant helix and determine a vector differential equation of the third order satisfied by the derivative of principal normal vector fields of a regular curve. In terms of solution, we determine the parametric representation of the slant-slant helix from the intrinsic equations.
 
 Finally, we present some examples of slant-slant helices by means of intrinsic equations.

Real Hypersurfaces in the Complex Quadric with Lie Invariant Structure Jacobi Operator

AbstractWe introduce the notion of Lie invariant structure Jacobi operators for real hypersurfaces in the complex quadric$Q^{m}=SO_{m+2}/SO_{m}SO_{2}$. The existence of invariant structure Jacobi operators implies that the unit normal vector field$N$becomes$\mathfrak{A}$-principal or$\mathfrak{A}$-isotropic. Then, according to each case, we give a complete classification of real hypersurfaces in$Q^{m}=SO_{m+2}/SO_{m}SO_{2}$with Lie invariant structure Jacobi operators.

New Approach to Slant Helix

A slant helix is a curve for which the principal normal vector field makes a constant angle with a fixed direction. In this study, we solve a system of linear ordinary differential equations involving an alternative moving frame, then determine the position vectors of slant helices through integration in Minkowski 3-space.

Ray mapping method for off-axis and non-paraxial freeform illumination lens design.

The ray mapping method for freeform illumination design is an easy and flexible method, but only in the paraxial regime does it result in surface normal fields that are directly integrable into continuous freeform surfaces that provide the desired illuminance distribution. A new mapping scheme is proposed to alter an initial source-target mapping via a symplectic flow of an equi-flux parametric coordinate system. The resulting mapping provides integrable surface normal vector fields for complex off-axis and non-paraxial illumination problems, as demonstrated by two freeform lens examples.

Real Hypersurfaces in the Complex Hyperbolic Quadric with Parallel Ricci Tensor

We introduce the notion of parallel Ricci tensor for real hypersurfaces in the complex hyperbolic quadric $${Q^m}^* = SO^{o}_{m,2}/SO_mSO_2$$ . According to the $$\mathfrak {A}$$ -principal or the $$\mathfrak {A}$$ -isotropic unit normal vector field N, we give a complete classification of real hypersurfaces in $${Q^m}^* = SO^{o}_{m,2}/SO_mSO_2$$ with Ricci parallelism.

Sharp Reilly-type inequalities for a class of elliptic operators on submanifolds

Let M be an n(>2)-dimensional closed orientable submanifold in an (n+p)-dimensional space form Rn+p(c). We obtain an optimal upper bound for the second eigenvalue of a class of elliptic operators on M defined by LTf=−div(T∇f), where T is a general symmetric, positive definite and divergence-free (1,1)-tensor on M. The upper bound is given in terms of an integration involving tr T and |HT|2, where tr T is the trace of the tensor T and HT=∑i=1nA(Tei,ei) is a normal vector field associated with T and the second fundamental form A of M. Furthermore, we give the sufficient and necessary conditions when the upper bound is attained. Our main theorem can be viewed as an extension of the famous “Reilly inequality”. The operator LT can be regarded as a natural generalization of the well-known operator Lr which is the linearized operator of the first variation of the (r+1)-th mean curvature for hypersurfaces in a space form. As applications of our main theorem, we generalize the results of Grosjean [17] and Li–Wang [20] in codimension one to arbitrary codimension.

Weakly Normal Basis Vector Fields in RKHS with an Application to Shape Newton Methods

We construct a space of vector fields that are normal to differentiable curves in the plane. Its basis functions are defined via saddle point variational problems in reproducing kernel Hilbert spac...