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 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 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.

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.

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.

Geodesics on the unit tangent bundle

2003 ◽  
Vol 133 (6) ◽  
pp. 1209-1229 ◽  
Author(s):  
J. Berndt ◽  
E. Boeckx ◽  
P. T. Nagy ◽  
L. Vanhecke
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Tangent Bundle ◽  
Unit Vector ◽  
Sphere Bundle ◽  
Sasaki Metric ◽  
Tangent Sphere Bundle ◽  
Tangent Sphere ◽  
Unit Tangent Bundle ◽  
Unit Tangent Sphere Bundle

A geodesic γ on the unit tangent sphere bundle T1M of a Riemannian manifold (M, g), equipped with the Sasaki metric gS, can be considered as a curve x on M together with a unit vector field V along it. We study the curves x. In particular, we investigate for which manifolds (M, g) all these curves have constant first curvature κ1 or have vanishing curvature κi for some i = 1, 2 or 3.

Real hypersurfaces in the complex hyperbolic quadric with normal Jacobi operator of Codazzi type

2021 ◽  
Author(s):  
Imsoon Jeong ◽  
Eunmi Pak ◽  
Young Jin Suh
Keyword(s):  
Jacobi Operator ◽  
Complete Classification ◽  
Real Hypersurfaces ◽  
Normal Vector ◽  
Normal Vector Field ◽  
Hyperbolic Quadric ◽  
Complex Hyperbolic Quadric ◽  
Unit Normal Vector Field ◽  
Codazzi Type ◽  
Normal Jacobi Operator

In this paper, we introduce the notion of normal Jacobi operator of Codazzi type for real hypersurfaces in the complex hyperbolic quadric [Formula: see text]. The normal Jacobi operator of Codazzi type implies that the unit normal vector field [Formula: see text] becomes [Formula: see text]-principal or [Formula: see text]-isotropic. Then according to each case, we give a complete classification of Hopf real hypersurfaces in [Formula: see text] with normal Jacobi operator of Codazzi type. The result of the classification shows that no such hypersurfaces exist.

scholarly journals Piecewise smooth reconstruction of normal vector field on digital data

2016 ◽  
Vol 35 (7) ◽  
pp. 157-167 ◽  
Author(s):  
David Coeurjolly ◽  
Marion Foare ◽  
Pierre Gueth ◽  
Jacques-Olivier Lachaud
Keyword(s):  
Vector Field ◽  
Piecewise Smooth ◽  
Digital Data ◽  
Normal Vector ◽  
Normal Vector Field
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