arXiv research

Characterizes magnetic unit vector fields on Lie groups.

Study biharmonic vector fields and unit vector fields on Riemannian manifolds.

We present a new equation with respect to a unit vector field on Riemannian manifold $M^n$ 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…

The bienergy of smooth maps between Riemannian manifolds, when restricted to unit vector fields, yields two different variational problems depending on whether one takes the full functional or just the vertical contribution. Their critical points, called biharmonic unit vector fields and biharmonic unit sections, form …

Minimal vector fields on oscillator groups studied, with specific conditions for minimality.

Sharp lower bound found for area of vector fields on spherical annuli.

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.

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…

Study finds the volume of unit vector fields on a punctured sphere and shows their images match minimally immersed Klein bottles.

Minimal surfaces in S3(2) linked to vector fields on punctured sphere.

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.

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 $J_{δ, σ}$ 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 $g_{δ, σ}$ on $T M$ which are …

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 …

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 …

We provide some examples of harmonic unit vector fields as normalized gradients of isoparametric functions from a K-contact geometry setting.

It is important in many applications 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 the paper is to provide an elementary proof of the existence and uniqueness of such an extension.

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 …

Outer billiards maps on foliated surfaces with specific vector fields.

In these short notes we characterize the loxodromic unit vector fields on antipodally punctured Euclidean spheres as the only ones achieving a lower bound for the volume functional depending on the Poincaré indexes around their singularities.

For $n\geq 1$, we exhibit a lower bound for the volume of a unit vector field on $\mathbb{S}^{2n+1}\backslash\{\pm p\}$ depending on the absolute values of its Poincaré indices around $\pm p$. We determine which vector fields achieve this volume, and discuss the idea of having multiple isolated singularities of arbitra…

New method constructs axial vector fields and defines quasi-local spin-angular momentum.

We obtain the explicit representation of Legendre surfaces in the unit $5$-sphere with harmonic mean curvature vector field, under the condition that the mean curvature function is constant along a certain special direction.

Minimal volume vector fields on surfaces via calibrations.

The paper "Minimal unit vector fields" by O. Gil-Medrano and E. Llinares-Fuster \cite{GilLli1}. is a seminal paper in the field that has been cited by many authors. It contains, however, a minor technical mistake in Theorem 14 that is important to fix. In this short note, we will provide a correction to that result.

Study properties of specific solitons on submanifolds with special vector fields.

We study 6-dimensional nearly Kahler manifolds admitting a Killing vector field of unit length. In the compact case it is shown that up to a finite cover there is only one geometry possible, that of the 3--symmetric space $S^3 \times S^3$.

New examples of harmonic unit vector fields on hyperbolic 3-space are constructed by exploiting the reduction of symmetry arising from the foliation by horospheres. This is compared and contrasted with the analogous construction in Euclidean 3-space, using a foliation by planes, which produces some new examples of harm…

The paper studies prescribed angle surfaces in Riemannian manifolds with torse-forming vector fields.

We present a boundary version of a theorem about solenoidal unit vector fields with minimum energy on a spherical domain of an odd dimensional Euclidean sphere.

In this paper, we define a certain "proportional volume property" for an unit vector field on a spherical domain in S3. We prove that the volume of these vector fields has an absolute minimum and this value is equal to the volume of the Hopf vector field. Some examples of such vector fields are given. We also study the…

Harmonic and minimal great circle fibrations have special Gauss maps.

The paper studies vector fields on manifolds and their embeddings into tangent bundles.

The paper classifies hypersurfaces in Riemannian manifolds with constant inner product and torse-forming axes.

We adapt the Newman-Penrose formalism in general relativity to the setting of three-dimensional Riemannian geometry, and prove the following results. Given a Riemannian 3-manifold without boundary and a smooth unit vector field ${\boldsymbol k}$ with geodesic flow, if an integral curve of ${\boldsymbol k}$ is hypersurf…

Given a compact manifold M, we prove that any bracket generating and invariant under multiplication on smooth functions family of vector fields on M generates the connected component of unit of the group Diff(M).

Injectivity result for light ray transform on Lorentzian manifolds.

Considering pseudo-Riemannian $g$-natural metrics on tangent bundles, we prove that the condition of being Ricci soliton is hereditary in the sense that a Ricci soliton structure on the tangent bundle gives rise to a Ricci soliton structure on the base manifold. Restricting ourselves to some class of pseudo-Riemannian …

Given a Hopf fibration of a round sphere by parallel great subspheres, we prove that the projection map to the base space is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. Similarly, given a Hopf fibration of a round sphere by parallel great circles, we view a unit…

Recently, the geodesibility of planar vector fields, which are algebrizable (differentiable in the sense of Lorch for some associative and commutative unital algebra), has been established. In this paper, we consider algebrizable three-dimensional vector fields, for which we give rectifications and Riemannian metrics u…

Study finds limits on curvature and shape of certain spacetimes.

We parametrize the space $\mathcal{Z}$ of Zygmund vector fields on the unit circle in terms of infinitesimal shear functions on the Farey tesselation. Then we express the Hilbert transform and the Fourier coefficients of the Zygmund vector fields in terms of the above parametrization by infinitesimal shear functions. F…

In this paper, we study spherical images of the modified orthogonal vector fields and Darboux vector of a regular curve which lies on the unit sphere in Euclidean 3-space.

The isotropic almost complex structures induce a Riemannian metric $g_{δ,σ}$ on TM, which are the generalized type of Sasakian metric. In this paper, the Levi-Civita connection of $g_{δ,σ}$ is calculated and the harmonicity of unit vector fields from $(M, g)$ to $(S(M), i*g_{δ,0})$ is investigated, where $i*g_{δ,0}$ is…

Geodesic vector fields on flat 3-manifolds are related to contact structures.

Classifies hypersurfaces with specific curvature properties in 4D space.