scholarly journals Constant angle spacelike surface in de Sitter space S₁³

2017 ◽  
Vol 35 (3) ◽  
pp. 79-93
Author(s):  
Tugba Mert ◽  
Baki Karlıga
Keyword(s):  
Vector Field ◽  
De Sitter Space ◽  
Normal Vector ◽  
Spacelike Surface ◽  
De Sitter ◽  
Unit Normal Vector ◽  
Normal Vector Field ◽  
Constant Angle ◽  
Unit Normal Vector Field ◽  
Unit Normal

In this paper; using the angle between unit normal vector field of surfaces and a fixed spacelike axis in R₁⁴, we develop two class of spacelike surface which are called constant timelike angle surfaces with timelike and spacelike axis in de Sitter space S₁³.

scholarly journals Extension of the unit normal vector field from a hypersurface

2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Roland Duduchava ◽  
Eugene Shargorodsky ◽  
George Tephnadze
Keyword(s):  
Vector Field ◽  
Existence And Uniqueness ◽  
Elementary Proof ◽  
Gradient Field ◽  
Normal Vector ◽  
Unit Normal Vector ◽  
Normal Vector Field ◽  
Unit Normal Vector Field ◽  
Outer Unit ◽  
Unit Normal

AbstractIn many applications it is important to be able to extend the (outer) unit normal vector field from a hypersurface to its neighborhood in such a way that the result is a unit gradient field. The aim of this paper is to provide an elementary proof of the existence and uniqueness of such an extension.

scholarly journals On compact minimal hypersurfaces in a sphere with constant scalar curvature

1980 ◽  
Vol 78 ◽  
pp. 177-188 ◽  
Author(s):  
Naoya Doi
Keyword(s):  
Vector Field ◽  
Constant Scalar Curvature ◽  
Arbitrary Point ◽  
Second Fundamental Form ◽  
Normal Vector ◽  
Unit Normal Vector ◽  
Covariant Differentiation ◽  
Normal Vector Field ◽  
Unit Normal Vector Field ◽  
Unit Normal

Let M be an n-dimensional hypersurface immersed in the (n + 1)-dimensional unit sphere Sn+1 with the standard metric by an immersion f. We denote by A the second fundamental form of the immersion / which is considered as a symmetric linear transformation of each tangent space TXM, i.e. for an arbitrary point x of M and the unit normal vector field ξ defined in a neighborhood of x, A is given by where is the covariant differentiation in Sn+i and Thus, A depends on the orientation of the unit normal vector field ξ and, in general, it is locally defined on M.

scholarly journals An integral formula for compact hypersurfaces in a Euclidean space and its applications

1992 ◽  
Vol 34 (3) ◽  
pp. 309-311 ◽  
Author(s):  
Sharief Deshmukh
Keyword(s):  
Vector Field ◽  
Euclidean Space ◽  
Smooth Function ◽  
Support Function ◽  
Integral Formula ◽  
Position Vector ◽  
Normal Vector ◽  
Unit Normal Vector ◽  
Normal Vector Field ◽  
Unit Normal Vector Field

Let M be a compact hypersurface in a Euclidena space ℝn+1. The support function p of M is the component of the position vector field of Min ℝn+1 along the unit normal vector field to M, which is a smooth function defined on M. Let S be the scalar curvature of M. The object of the present paper is to prove the following theorems.

Le problème de Neumann pour certaines équations du type de Monge-Ampère sur une variété riemannienne

2000 ◽  
Vol 52 (4) ◽  
pp. 757-788
Author(s):  
Abdellah Hanani
Keyword(s):  
Elliptic Partial Differential Equation ◽  
Positive Function ◽  
Neumann Condition ◽  
Normal Vector ◽  
Unit Normal Vector ◽  
Normal Vector Field ◽  
Nonlinear Elliptic ◽  
Unit Normal Vector Field ◽  
Monge Ampere ◽  
Strictly Positive Function

AbstractLet (Mn, g) be a strictly convex riemannian manifold with C∞ boundary. We prove the existence of classical solution for the nonlinear elliptic partial differential equation of Monge-Ampère: det in M with a Neumann condition on the boundary of the form , where is an everywhere strictly positive function satisfying some assumptions, ν stands for the unit normal vector field and is a non-decreasing function in u.

A new classification on parallel Ricci tensor for real hypersurfaces in the complex quadric

2020 ◽  
pp. 1-23
Author(s):  
Hyunjin Lee ◽  
Young Jin Suh
Keyword(s):  
Ricci Tensor ◽  
Real Hypersurfaces ◽  
Normal Vector ◽  
Unit Normal Vector ◽  
New Classification ◽  
Normal Vector Field ◽  
Complex Quadric ◽  
Unit Normal Vector Field ◽  
Parallel Ricci Tensor

First we introduce the notion of parallel Ricci tensor ${\nabla }\mathrm {Ric}=0$ for real hypersurfaces in the complex quadric Q m  = SOm+2/SO m SO2 and show that the unit normal vector field N is singular. Next we give a new classification of real hypersurfaces in the complex quadric Q m  = SOm+2/SO m SO2 with parallel Ricci tensor.

scholarly journals Focal representation of k-slant Helices in Em+1

2015 ◽  
Vol 7 (2) ◽  
pp. 200-209 ◽  
Author(s):  
Günay Öztürk ◽  
Betül Bulca ◽  
Bengü Bayram ◽  
Kadri Arslan
Keyword(s):  
Normal Vector ◽  
Unit Normal Vector ◽  
Regular Curve ◽  
Fixed Direction ◽  
Constant Angle ◽  
Slant Helix ◽  
Unit Normal

Abstract The focal representation of a generic regular curve γ in Em+1 consists of the centers of the osculating hyperplanes. A k-slant helix γ in Em+1 is a (generic) regular curve whose unit normal vector Vk makes a constant angle with a fixed direction in Em+1. In the present paper we proved that if γ is a k-slant helix in Em+1, then the focal representation Cγ of γ in Em+1 is an (m− k + 2)-slant helix in Em+1.

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

2019 ◽  
Vol 12 (06) ◽  
pp. 2040009
Author(s):  
Hatce Kusak Samanci ◽  
Ayhan Yildiz
Keyword(s):  
Vector Field ◽  
Normal Vector ◽  
Normal Vector Field ◽  
Spacelike Curve ◽  
Bishop Frame ◽  
Constant Angle ◽  
Slant Helix ◽  
Constant Direction ◽  
Definition Of

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.

scholarly journals Integral Formulas for a Foliation with a Unit Normal Vector Field

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1764
Author(s):  
Vladimir Rovenski
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Shape Operator ◽  
Unit Vector ◽  
Normal Vector ◽  
Normal Vector Field ◽  
Codimension One ◽  
Integral Formulas ◽  
Newton Transformations ◽  
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.

scholarly journals L p -trace-free generalized Korn inequalities for incompatible tensor fields in three space dimensions

2021 ◽  
pp. 1-32
Author(s):  
Peter Lewintan ◽  
Patrizio Neff
Keyword(s):  
Normal Vector ◽  
Tensor Fields ◽  
Unit Normal Vector ◽  
Normal Vector Field ◽  
Bounded Lipschitz Domain ◽  
Image Position ◽  
Unit Normal Vector Field ◽  
Outward Unit ◽  
Three Space ◽  
Outward Unit Normal Vector

For $1< p<\infty$ we prove an $L^{p}$ -version of the generalized trace-free Korn inequality for incompatible tensor fields $P$ in $W^{1,p}_0(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$ . More precisely, let $\Omega \subset \mathbb {R}^{3}$ be a bounded Lipschitz domain. Then there exists a constant $c>0$ such that \[ \lVert{ P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\leq c\,\left(\lVert{\operatorname{dev} \operatorname{sym} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})} + \lVert{ \operatorname{dev} \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\right) \] holds for all tensor fields $P\in W^{1,p}_0(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$ , i.e., for all $P\in W^{1,p} (\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$ with vanishing tangential trace $P\times \nu =0$ on $\partial \Omega$ where $\nu$ denotes the outward unit normal vector field to $\partial \Omega$ and $\operatorname {dev} P : = P -\frac 13 \operatorname {tr}(P) {\cdot } {\mathbb {1}}$ denotes the deviatoric (trace-free) part of $P$ . We also show the norm equivalence \begin{align*} &\lVert{ P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}+\lVert{ \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\\ &\quad\leq c\,\left(\lVert{P}\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})} + \lVert{ \operatorname{dev} \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\right) \end{align*} for tensor fields $P\in W^{1,p}(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$ . These estimates also hold true for tensor fields with vanishing tangential trace only on a relatively open (non-empty) subset $\Gamma \subseteq \partial \Omega$ of the boundary.

Real hypersurfaces in the complex quadric with Killing normal Jacobi operator

2018 ◽  
Vol 149 (2) ◽  
pp. 279-296 ◽  
Author(s):  
Young Jin Suh
Keyword(s):  
Jacobi Operator ◽  
Complete Classification ◽  
Real Hypersurfaces ◽  
Normal Vector ◽  
Unit Normal Vector ◽  
Normal Vector Field ◽  
Complex Quadric ◽  
Unit Normal Vector Field ◽  
Normal Jacobi Operator

AbstractWe introduce the notion of Killing normal Jacobi operator for real hypersurfaces in the complex quadricQm=SOm+2/SOmSO2. The Killing normal Jacobi operator implies that the unit normal vector fieldNbecomes 𝔄-principal or 𝔄-isotropic. Then according to each case, we give a complete classification of real hypersurfaces inQm=SOm+2/SOmSO2with Killing normal Jacobi operator.

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