arXiv research

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

The paper studies integrability of isotropic almost complex structures and harmonic unit vector fields.

The study proves biharmonic unit sections on 2-tori are always harmonic and exists in each homotopy class.

Study classifies harmonic vector fields on 3-manifolds.

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

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

Harmonic and minimal great circle fibrations have special Gauss maps.

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.

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…

Study on $H$-contact structures on Riemannian 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…

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…

Paper approximates fractional harmonic maps with numerical methods.

Harmonic unit normal sections studied for Grassmannians induced by cross products.

An almost contact metric structure is parametrized by a section of an associated homogeneous fibre bundle, and conditions for this to be a harmonic section, and a harmonic map, are studied. These involve the characteristic vector field, and the almost complex structure in the contact subbundle. Several examples are giv…

Harmonic basis vector fields on surfaces

The study explores harmonic vector fields on a specific type of Riemannian Lie group.

The theory of harmonic vector fields on Riemannian manifolds is generalised to pseudo-Riemannian manifolds. Harmonic conformal gradient fields on pseudo-Euclidean hyperquadrics are classified up to congruence, as are harmonic Killing fields on pseudo-Riemannian quadrics. A para-Kaehler twisted anti-isometry is used to …

The paper characterizes vector fields as interpolating sesqui-harmonic maps on Riemannian manifolds.

Study on vector fields on Lie groups reveals surprising algebraic coincidences.

Study on biharmonic and interpolating sesqui-harmonic vector fields on para-Kähler--Norden manifolds.

A vector field s on a Riemannian manifold M is said to be harmonic if there exists a member of a 2-parameter family of generalised Cheeger-Gromoll metrics on TM with respect to which s is a harmonic section. If M is a simply-connected non-flat space form other than the 2-sphere, examples are obtained of conformal vecto…

We consider four dimensional lie groups equipped with left invariant Lorentzian Einstein metrics, and determine the harmonicity properties of vector fields on these spaces. In some cases, all these vector fields are critical points for the energy functional restricted to vector fields. We also classify vector fields de…

Characterizes magnetic unit vector fields on Lie groups.

The absence of interesting harmonic sections for the Sasaki and Cheeger-Gromoll metrics has led to the consideration of alternatives, for example in the form of a two-parameter family of natural metrics shown to relax existence conditions for harmonicity. This article investigates harmonic Killing vector fields, proves…

Study on points where random spherical harmonic nodal set meets tangent vector field.

The paper classifies Ricci solitons and studies harmonic vector fields on a specific Thurston geometry.

The paper proves nonexistence of harmonic and bi-harmonic maps under specific conditions.

New definitions and properties of harmonic vector fields on Finsler manifolds.

Paper proves extension of unit normal vector field from a hypersurface.

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…

We study harmonic sections of a Riemannian vector bundle whose total space is equipped with a 2-parameter family of metrics which includes both the Sasaki and Cheeger-Gromoll metrics. This enables the theory of harmonic unit sections to be extended to bundles with non-zero Euler class.

This article studies the harmonicity of vector fields on Riemannian manifolds, viewed as maps into the tangent bundle equipped with a family of Riemannian metrics. Geometric and topological rigidity conditions are obtained, especially for surfaces and vector fields of constant norm, and existence is proved on two-tori.…

The study characterizes harmonic spaces and their radial eigen-functions and vector fields.

Lower bounds on unit vector fields' volume using Poincaré indices.

Study surfaces in Half-Pipe space and vector fields on hyperbolic plane.

Study geometrical properties of oscillator group with a Lorentzian metric.

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

Characterizes loxodromic unit vector fields on punctured spheres.

Our aim in this paper is to investigate some geometrical properties of Berger Spheres i.e. homogeneous Ricci solitons and harmonicity properties of invariant vector fields. We determine all vector fields which are critical points for the energy functional restricted to vector fields. We also see that do not exist any v…

Study of vortex interactions in Ginzburg-Landau models on 2D Riemannian manifolds.

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.

A submanifold $M^m$ of a Euclidean space $R^{m+p}$ is said to have harmonic mean curvature vector field if $Δ\vec{H}=0$, where $\vec{H}$ is the mean curvature vector field of $M\hookrightarrow R^{m+p}$ and $Δ$ is the rough Laplacian on $M$. There is a conjecture named after Bangyen Chen which states that submanifolds o…

Let $(M,g)$ be a Riemannian manifold. When $M$ is compact and the tangent bundle $TM$ is equipped with the Sasaki metric $g^s$, the only vector fields which define harmonic maps from $(M,g)$ to $(TM,g^s)$, are the parallel ones. The Sasaki metric, and other well known Riemannian metrics on $TM$, are particular examples…