Paper proves extension of unit normal vector field from a hypersurface.
problem Need to extend unit normal vector field from a hypersurface.
method Provides an elementary proof of existence and uniqueness of such an extension.
result Elementary proof of existence and uniqueness of unit gradient field extension.
We present a new equation with respect to a unit vector field on Riemannian manifold Mn such that its solution defines a totally geodesic submanifold in the unit tangent bundle with Sasaki metric and apply it to some classes of unit vector fields. We introduce a class of covariantly normal unit vector fields and pro…
We provide some examples of harmonic unit vector fields as normalized gradients of isoparametric functions from a K-contact geometry setting.
We consider a unit normal vector field of (local) hyperfoliation on a given Riemannian manifold as a submanifold in the unit tangent bundle with Sasaki metric. We give an explicit expression of the second fundamental form for this submanifold and a rather simple condition its totally geodesic property in the case of a …
New equations connect unit Killing vectors to initial data.
problem Characterizing initial data for Einstein vacuum with unit Killing vectors.
method Developed new equations (uKID) by eliminating scaling and using propagation identity.
result Found equations that are finite type and characterize unit normalized Killing vectors.
We prove that the Hopf vector field is a unique one among geodesic covariantly normal unit vector fields on spheres such that the submanifold generated by the field is totally geodesic in the unit tangent bundle with Sasaki metric. As application, we give a new proof of stability (instability) of the Hopf vector field …
New method for mesh denoising using TGV of normal vector field.
problem Improving mesh quality by removing noise.
method Proposes a novel TGV formulation for normal vector fields on triangular meshes.
result New method outperforms existing techniques in mesh denoising experiments.
The paper studies prescribed angle surfaces in Riemannian manifolds with torse-forming vector fields.
problem Characterizing surfaces with prescribed angles in Riemannian geometry.
method Introducing and analyzing prescribed angle hypersurfaces associated with torse-forming vector fields.
result Classification of prescribed angle surfaces in 3D Riemannian manifolds.
Outer billiards maps on foliated surfaces with specific vector fields.
problem Characterizing vector fields that induce outer billiards maps on foliated surfaces.
method Analyzing necessary and sufficient conditions for foliation of the exterior of a hypersurface.
result Explicit periodic and unbounded orbits in a specific outer billiard map.
The paper classifies hypersurfaces in Riemannian manifolds with constant inner product and torse-forming axes.
problem Understanding hypersurfaces in Riemannian manifolds with specific geometric properties.
method Analyzing hypersurfaces with constant inner product and torse-forming axes.
result Classification of hypersurfaces with torse-forming axes.
Characterizes magnetic unit vector fields on Lie groups.
problem Classifying magnetic unit vector fields on Lie groups.
method Characterization through critical points of Landau Hall and Dirichlet energy functionals.
result Classification of all magnetic left invariant unit vector fields on 3-dimensional Lie groups.
In this article a relation between curvature functionals for surfaces in the Euclidean space and area functionals in relative differential geometry will be given. Relative differential geometry can be described as the geometry of surfaces in the affine space, endowed with a distinguished "relative normal vector field" …
We give a complete description of all hypersurfaces of the product spaces $\Sf^n\times \R$ and $\Hy^n\times \R$ that have flat normal bundle when regarded as submanifolds with codimension two of the underlying flat spaces $\R^{n+2}\supset \Sf^n\times \R$ and $\Le^{n+2}\supset \Hy^n\times \R$. We prove that any such hyp…
Study biharmonic vector fields and unit vector fields on Riemannian manifolds.
problem Determine the equivalence of biharmonicity and harmonicity for vector fields and unit vector fields on Riemannian manifolds.
method Analyze biharmonic vector fields and unit vector fields on (M,g) with pseudo-Riemannian g-natural metrics on TM and T1M. result Contrary to Sasaki metric, biharmonicity and harmonicity are not equivalent for large classes of g-natural metrics on TM. The study proves biharmonic unit sections on 2-tori are always harmonic and exists in each homotopy class.
problem Characterizing biharmonic unit vector fields and sections on 2-tori.
method Analyzing variational problems for unit vector fields under conformal metrics, proving properties through homotopy classes.
result Biharmonic unit sections on 2-tori are always harmonic and exist in each homotopy class.
In 1963, K.P.Grotemeyer proved an interesting variant of the Gauss-Bonnet Theorem. Let M be an oriented closed surface in the Euclidean space R^3 with Euler characteristic χ(M), Gauss curvature G and unit normal vector field n. Grotemeyer's identity replaces the Gauss-Bonnet integrand G by the normal moment <a,n>^2G, w…
Harmonic unit normal sections studied for Grassmannians induced by cross products.
problem Energy of maps assigning unit vectors to subspaces of Grassmannians.
method Analyzing cross products to induce harmonic sections into sphere bundles.
result All unit normal sections of Grassmannians associated with cross products are harmonic.
Minimal vector fields on oscillator groups studied, with specific conditions for minimality.
problem Characterizing minimal left-invariant unit vector fields on oscillator groups.
method Analyzing structure constants and harmonic maps into the unit tangent bundle.
result Minimal vector fields defined by specific conditions on oscillator groups.
Lower bounds on unit vector fields' volume using Poincaré indices.
problem Finding volume bounds for unit vector fields on spheres with punctures.
method Using Poincaré indices to establish lower bounds and identify achieving fields.
result Identified vector fields achieving the determined volume bounds.
The paper studies vector fields on manifolds and their embeddings into tangent bundles.
problem Characterizing and understanding vector fields on manifolds and their embeddings.
method Forming Sasaki metrics and studying embeddings of cross sections defined by vector fields.
result Minimal unit vector fields on spheres are related to contact structures.
Sharp lower bound found for area of vector fields on spherical annuli.
problem Finding the minimum area of unit vector fields on spherical annuli.
method Established a sharp lower bound through mathematical analysis.
result Sharp lower bound for the area of unit vector fields on spherical annuli.
Characterizes loxodromic unit vector fields on punctured spheres.
problem Finding vector fields with a lower bound on volume functional.
method Characterization based on Poincaré indexes.
result Only loxodromic unit vector fields achieve the lower bound.
We give a complete list of those left invariant unit vector fields on three-dimensional Lie groups with the left-invariant metric that generate a totally geodesic submanifold in the unit tangent bundle of a group with the Sasaki metric. As a result, each class of three-dimensional Lie groups admits the totally geodesic…
Study finds a minimum volume for vector fields on a punctured sphere.
problem Finding the minimum volume of unit vector fields on a punctured sphere.
method Analyzes the volume of vector fields tangent to an antipodally punctured unit 2-sphere.
result Provides a lower bound for the volume of unit vector fields.
We present an explicit formula for the mean curvature of a unit vector field on a Riemannian manifold, using a special but natural frame. As applications, we treat some known and new examples of minimal unit vector fields. We also give an example of a vector field of constant mean curvature on the Lobachevsky (n+1) s…
The paper classifies solitons in a curved product space.
problem Classifying solitons in a curved product space.
method Examined vector fields tangent to fibers and rotations, classified solitons under specific symmetries.
result A classification of solitons in s2imesR under certain symmetries. Study finds the volume of unit vector fields on a punctured sphere and shows their images match minimally immersed Klein bottles.
problem Finding the volume of unit vector fields on a punctured sphere.
method Analyzes the volume of unit vector fields on an antipodally punctured unit 2-sphere and shows their images coincide with minimally immersed Klein bottles.
result The images of minimizing vector fields on the punctured sphere match those of minimally immersed Klein bottles.
The paper defines and proves the existence of curves in Riemannian manifolds with prescribed angles to torse-forming vector fields.
problem Existence of curves with prescribed angles to torse-forming vector fields in Riemannian manifolds.
method Introducing the notion of a prescribed angle curve and proving its existence for torse-forming vector fields.
result Existence of prescribed angle curves in Riemannian manifolds associated with torse-forming vector fields.
The paper classifies and describes translators in SL(2,R) under specific symmetry conditions.
problem Classifying translators in SL(2,R) under invariant symmetry groups. method Analyzing translators invariant by one-parameter groups of isometries, using Iwasawa decomposition and Killing vector fields.
result Explicit parametrizations of translators are obtained for some cases.
The paper derives new Gauss-Bonnet formulas for frontal bundles over surfaces with boundary.
problem Deriving new formulas for coherent tangent bundles over surfaces with boundary.
method Defining frontal bundles and applying Gauss-Bonnet theorems to derive formulas.
result Four new Gauss-Bonnet type formulas for frontal bundles are derived.
Study space-like surfaces in Robertson-Walker spacetimes with specific geometric conditions.
problem Characterize space-like surfaces in Robertson-Walker spacetimes with given geometric conditions.
method Investigate surfaces satisfying specific conditions on tangential and normal parts of the unit vector field, using shape operators and minimal surfaces.
result Classification theorem and parametrizations of space-like class A surfaces in L14(f,0). Minimal surfaces in S3(2) linked to vector fields on punctured sphere.
problem Connecting minimal surfaces in S3(2) to vector fields on a punctured sphere.
method Established a correspondence between minimal surfaces and area-minimizing vector fields.
result Stability relation for Lawson cylinders in S3(2).
In this paper, we consider a regular curve on an oriented surface in Euclidean 3-space with the Darboux frame {T,V,U} along the curve, where T is the unit tangent vector field of the curve, U is the surface normal restricted to the curve and $\mathsf{V}=\mathsf{U}\ti…
We give a full geometrical description of local totally geodesic unit vector field on Riemannian 2-manifold, considering the field as a local imbedding of the manifold into its unit tangent bundle with the Sasaki metric.
Study classifies harmonic vector fields on 3-manifolds.
problem Classifying harmonic unit vector fields on 3-manifolds.
method Investigates under mild curvature assumptions, classifying vector fields and manifolds.
result Classifies both vector fields and manifolds supporting them.
Study spherical images of modified vector fields on a unit sphere.
problem Understanding spherical images of modified vector fields.
method Analysis of spherical images using modified orthogonal vector fields and Darboux vector.
result Characterization of spherical indicatrices with modified orthogonal frame.
A constant angle surface in Minkowski space is a spacelike surface whose unit normal vector field makes a constant hyperbolic angle with a fixed timelike vector. In this work we study and classify these surfaces. In particular, we show that they are flat. Next we prove that a tangent developable surface (resp. cylinder…
The paper studies properties of RCD(K,N) spaces and their boundaries.
problem Understanding the boundary structure and unit normal on RCD(K,N) spaces. method Proves concentration of boundary measure, discusses localization of unit normal, and develops tools for perimeter minimizers.
result Proves that the boundary measure of sets with finite perimeter is concentrated on the n-regular set Rn. Along cuspidal edge singularities on a given surface in Euclidean 3-space, which can be parametrized by a regular space curve, a unit normal vector field ν is well-defined as a smooth vector field of the surface. A cuspidal edge singular point is called generic if the osculating plane of the cuspidal edge (as a regul…
The stability of the 3-dimensional Hopf vector field, as a harmonic section of the unit tangent bundle, is viewed from a number of different angles. The spectrum of the vertical Jacobi operator is computed, and compared with that of the Jacobi operator of the identity map on the 3-sphere. The variational behaviour of t…
Aguilar introduced isotropic almost complex structures Jδ,σ on the tangent bundle of a Riemannian manifold (M,g). In this paper, some results will be obtained on the integrability of these structures. These structures with the Liouville 1-form define a class of Riemannian metrics gδ,σ on TM which are …
We study the geometrical properties of a unit vector field on a Riemannian 2-manifold, considering the field as a local imbedding of the manifold into its tangent sphere bundle with the Sasaki metric. For the case of constant curvature K, we give a description of the totally geodesic unit vector fields for K=0 and K=1 …
Study pinches intrinsic and normal curvatures of minimal surfaces in a sphere.
problem Pinching constraints on intrinsic and normal curvatures of minimal surfaces.
method Established orthonormal frame field, derived property K+KN=1, used to pinch curvatures. result Pinched constraints on intrinsic and normal curvatures of minimal surfaces.
In this work, we study plane and spherical curves in Euclidean and Lorentz-Minkowski 3-spaces by employing rotation minimizing (RM) frames. By conveniently writing the curvature and torsion for a curve on a sphere, we show how to find the angle between the principal normal and an RM vector field for spherical curves. L…
The paper explores biconservative surfaces in a 4D sphere, finding a unique family of non-isometric surfaces.
problem Characterizing biconservative surfaces with a parallel normalized mean curvature vector field in a 4D sphere.
method Analyzes existence and uniqueness, derives local parametrization.
result A 2-parameter family of non-isometric biconservative surfaces in a 4D sphere.
New families of translating solitons found in hyperbolic space.
problem Classifying and proving the existence/non-existence of ξ-translators. method Using Killing vector fields and mean curvature conditions.
result Existence of a new family of grim reapers.
The paper studies polar normalizations of skew ruled surfaces in 3D space.
problem Understanding the properties and invariants of polar normalized skew ruled surfaces.
method Determination of invariants and analysis of Tchebychev and support vector fields.
result Special polar normalizations lead to degenerate curves.
New method constructs axial vector fields and defines quasi-local spin-angular momentum.
problem Constructing axial vector fields on Riemannian two-spheres.
method Using centre-of-mass unit sphere reference systems and Lie-propagated unit sphere reference systems.
result Constructive definition of quasi-local spin-angular momentum and balance relations.