arXiv research

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

Minimal volume vector fields on surfaces via calibrations.

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 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…

Minimal vector fields on a 2-sphere with varying volumes are discovered.

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

Study finds a minimum volume for vector fields on a punctured sphere.

Harmonic and minimal great circle fibrations have special Gauss maps.

We study invariant submanifolds of manifolds endowed with a normal or complex metric contact pair with decomposable endomorphism field $φ$. For the normal case, we prove that a $φ$-invariant submanifold tangent to a Reeb vector field and orthogonal to the other one is minimal. For a $φ$-invariant submanifold $N$ everyw…

The study identifies all possible vector field structures on specific 2D shapes.

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…

New proof of minimal vector fields on spheres using calibrations.

We prove that a normal vector field along a curve in R3 is rotation minimizing (RM) if and only if it is parallel respect to the normal connection. This allows us to generalize all the results of RM vectors and frames to curves immersed in Riemannian manifolds.

In this paper we proof that the Holomorphic angle for compact minimal surfaces in the sphere $S^5$ with constant Contact angle and with a parallel normal vector field must be constant.

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 "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.

In this paper, we propose one index $i_1(f)-i_2(f)$ which measures how well-behaved a given finitely determined multigerm $f: (\mathbb{K}^n,S)\to (\mathbb{K}^p,0)$ $(n\le p)$ of corank at most one is from the viewpoint of liftable vector fields; and we answer the following problems when the index indicates that the giv…

New concept of $V$-minimality applied to Kaehler and non-Kaehler manifolds.

Rotation minimizing vector fields and frames were introduced by Bishop as an alternative to the Frenet frame. They are used in CAGD because they can be defined even the curvature vanishes. Nevertheless, many other geometric properties have not been studied. In the present paper, RM vector fields along a curve immersed …

Study on existence and structure of P-area surfaces in Heisenberg group.

New method learns vector fields from noisy time series data.

In this article, we investigate the stability of leaves of minimal foliations of arbitrary codimension. We also study relations between Jacobi fields and vector fields which preserves a foliation and we use these results to Killing fields.

Minimal graphs over non-compact domains in 3-manifolds solved with estimates and uniqueness results.

The study examines vector fields with integer singularities in 3D balls.

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…

Minimal surfaces in spheres have unique energy properties.

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

A function that optimally aligns a timelike vector field with its gradients

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…

The purpose of this note is to establish the following theorem: Let N be a Kahler manifold, L be a compact oriented immersed minimal Lagrangian submanifold in N and V be a holomorphic vector field in a neighbourhood of L in N. Let div(V) be the (complex) divergence of V. Then the integral of div(V) over L is 0. Vice ve…

Braided vector fields on spatial subdomains homeomorphic to the cylinder play a crucial role in applications such as solar and plasma physics, relativistic astrophysics, fluid and vortex dynamics, elasticity, and bio-elasticity. Often the vector field's topology -- the entanglement of its field lines -- is non-trivial,…

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

New method AM learns optimal vector fields for entire distribution sequences, matching OT.

The paper defines flows on $\mathbb{Z}$-graded manifolds and proves unique maximal flows for vector fields.

Solves Jenkins-Serrin problem in 3-manifolds with Killing vector fields.

Given a triangulated region in the complex plane, a discrete vector field $Y$ assigns a vector $Y_i\in \mathbb{C}$ to every vertex. We call such a vector field holomorphic if it defines an infinitesimal deformation of the triangulation that preserves length cross ratios. We show that each holomorphic vector field can b…

In this paper, we completely classify almost Yamabe solitons on hypersurfaces in Euclidean spaces arisen from the position vector field. Some results of almost Yamabe solitons with a concurrent vector field and almost Yamabe solitons on submanifolds in Riemannian manifolds equipped with a concurrent vector field are al…

A morph between two Riemannian $n$-manifolds is an isotopy between them together with the set of all intermediate manifolds equipped with Riemannian metrics. We propose measures of the distortion produced by some classes of morphs and diffeomorphisms between two isotopic Riemannian $n$-manifolds and, with respect to th…

Study foliations on Riemannian manifolds with specific vector fields, focusing on geometric properties.

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…

It is of interest to study supergravity solutions preserving a non-minimal fraction of supersymmetries. A necessary condition for supersymmetry to be preserved is that the spacetime admits a Killing spinor and hence a null or timelike Killing vector field. Any spacetime admitting a covariantly constant null vector fiel…

In this article we perform a computational study of Polyrakis algorithms presented in [12,13]. These algorithms are used for the determination of the vector sublattice and the minimal lattice-subspace generated by a finite set of positive vectors of R^k. The study demonstrates that our findings can be very useful in th…

We show that $φ$-invariant submanifolds of metric contact pairs with orthogonal characteristic foliations make constant angles with the Reeb vector fields. Our main result is that for the normal case such submanifolds of dimension at least $2$ are all minimal. We prove that an odd-dimensional $φ$-invariant submanifold …

New formula shows how causal vectors relate to mass-minimizing data.

Adam optimizer converges to zeros of a new vector field, not just gradient zeros.

Develops a new exponential map for time-varying vector fields.

An integrated approach to Lie derivatives of spinors, spinor connections and the gravitational field is presented, in the context of a previously proposed, partly original formulation of a theory of Einstein-Carta-Maxwell-Dirac fields based on "minimal geometric data": all the needed underlying structure is geometrical…