arXiv research

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 …

Study classifies harmonic vector fields on 3-manifolds.

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…

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…

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

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.

Harmonic and minimal great circle fibrations have special Gauss maps.

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

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

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.

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

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

Study geometrical properties of oscillator group with a Lorentzian metric.

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 biharmonic vector fields and unit vector fields on Riemannian manifolds.

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

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.

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…

The concept of the Ricci soliton was introduced by Hamilton. Ricci soliton is defined by vector field and it's a natural generalization of Einstein metric. We have shown earlier that the vector field of Ricci soliton is an infinitesimal harmonic transformation. In our paper, we survey Ricci solitons geometry as an appl…

We introduce and study $H$-paracontact metric manifolds, that is, paracontact metric manifolds whose Reeb vector field $ξ$ is harmonic. We prove that they are characterized by the condition that $ξ$ is a Ricci eigenvector. We then investigate how harmonicity of the Reeb vector field $ξ$ of a paracontact metric manifold…

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…

Harmonic forms and Rumin complex linked on Sasakian manifolds.

Study harmonicity on tangent bundles with a specific metric.

Proves openness of balanced HKT cone and studies hyperholomorphic vector fields.

Study proves density of harmonic fields in continuous quaternion fields.

The paper connects two descriptions of Teichmüller space tangent spaces using harmonic vector fields.

The paper studies a new submersion type with specific conditions.

The study classifies gradient Ricci solitons with specific vector fields.

The paper examines $HM$-tensional and $HS$-tensional maps between Riemannian manifolds.

Study on $H$-contact structures on Riemannian manifolds.

Classifies sections of Riemannian bundles on Lie groups.

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…

On a Hermitian manifold we construct a symmetric $(1,1)$- tensor $H$ using the torsion and the curvature of the Chern connection. On a compact balanced Hermitian manifold we find necessary and sufficient conditions in terms of the tensor $H$ for a harmonic $1$-form to be analytic and for an analytic $1$-form to be harm…

Study geometry on $G_2$-manifolds, focusing on $Spin(7)$ and $G_2$.

The harmonic action functional allows a natural generalisation to semi-Riemannian supergeometry, referred to as superharmonic action, which resembles the supersymmetric sigma models studied in high energy physics. We show that Killing vector fields are infinitesimal supersymmetries of the superharmonic action and prove…

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…

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 paper introduces and characterizes almost ω-Bach solitons on various product manifolds.