arXiv research

We study a 1-form which can be given by a vector in a conformally invariant way. We then study conformally invariant functionals associated to a ``Y-diagram'' on the space of knots which are made from the 1-form.

Let $M^n$ be an $n$-dimensional umbilic-free hypersurface in the $(n+1)$-dimensional Lorentzian space form $M^{n+1}_1(c)$. Three basic invariants of $M^n$ under the conformal transformation group of $M^{n+1}_1(c)$ are a $1$-form $C$, called conformal $1$-form, a symmetric $(0,2)$ tensor $B$, called conformal second fun…

The paper explores how magnetic systems' spectra can identify metrics and 1-forms.

The paper defines vector 1-forms on Finsler manifolds and constructs connections.

The paper classifies Landsberg metrics under specific conditions.

In this paper, the Douglas curvature of (α,β)-metrics, a special class of Finsler metrics defined by a Riemannian metric αand a 1-form β, is studied. These metrics with vanishing Douglas curvature in dimension n\geq3 are classified by using a new class of metrical deformations called β-deformations. The result shows th…

The study classifies gradient Ricci solitons with specific vector fields.

We define a conformal reference frame, i.e., a special projection of the six-dimensional sky bundle of a Lorentzian manifold (or the five-dimensional twistor space) to a three-dimensional manifold. We construct an example, a conformal compactification, for Minkowski space. Based on the complex structure on the skies, w…

This paper classifies LCSKT almost abelian Lie algebras in 6 dimensions.

It is shown that an HKT-space with closed parallel potential 1-form has $D(2,1;-1)$-symmetry. Every locally conformally hyperkähler manifold generates this type of geometry. The HKT-spaces with closed parallel potential 1-form arising in this way are characterized by their symmetries and an inhomogeneous cubic conditio…

Study shows that dimension of Dolbeault harmonic forms is not always equal to B- on certain 4-manifolds.

The paper studies local normal forms of singular contact forms and primitive 1-forms.

The paper identifies magnetic ground states and their role in determining the conformal class of a surface.

We study transverse conformal Killing forms on foliations and prove a Gallot-Meyer theorem for foliations. Moreover, we show that on a foliation with $C$-positive normal curvature, if there is a closed basic 1-form $φ$ such that $Δ_Bφ=qCφ$, then the foliation is transversally isometric to the quotient of a $q$-sphere.

5D SCFTs can have confining vacua with strings and unbroken symmetries.

The abstract discusses conjectures about metrics on complex manifolds.

A conformal metric $g$ with constant curvature one and finite conical singularities on a compact Riemann surface $Σ$ can be thought of as the pullback of the standard metric on the 2-sphere by a multi-valued locally univalent meromorphic function $f$ on $Σ\backslash \{{\rm singularities}\}$, called the {\it developing …

The connected components of the zero set of any conformal vector field $v$, in a pseudo-Riemannian manifold $(M,g)$ of arbitrary signature, are of two types, which may be called `essential' and `nonessential'. The former consist of points at which $v$ is essential, that is, cannot be turned into a Killing field by a lo…

Constructs metrics on Riemann surfaces with singularities.

Study on Lee classes of complex surfaces, proving connectedness and bounds.

We investigate what we call a conformal $β$ - change in Finsler spaces, namely $L(x,y)\to ~^{\ast}L(x,y)=e^{σ(x)}L(x,y)+β(x,y)$ where~$σ~$ is a function of $x~ only ~and ~ β(x, y)$ is a given 1- form. This change generalizes various types of changes: conformal changes, Randers changes and $β$ - changes. Under this c…

Yamabe solitons defined on specific Sasaki-like manifolds.

We study conditions for which the mapping torus of a 6-manifold endowed with an $SU(3)$-structure is a locally conformal calibrated $G_2$-manifold, that is, a 7-manifold endowed with a $G_2$-structure $\varphi$ such that $d \varphi = - θ\wedge \varphi$ for a closed non-vanishing 1-form $θ$. Moreover, we show that if $(…

We establish a Hard Lefschetz Theorem for the de Rham cohomology of compact Vaisman manifolds. A similar result is proved for the basic cohomology with respect to the Lee vector field. Motivated by these results, we introduce the notions of a Lefschetz and of a basic Lefschetz locally conformal symplectic (l.c.s.) mani…

The study proves that conformal Killing vector fields on manifolds with positive Ricci curvature are non-trivial.

Locally conformally product Lie algebras are characterized and constructed.

We present some examples of locally conformal symplectic structures of the first kind on compact nilmanifolds which do not admit Vaisman metrics. One of these examples does not admit locally conformal Kähler metrics and all the structures come from left-invariant locally conformal symplectic structures on the correspon…

The paper describes conformal structures and Pfaffian systems for rolling surfaces.

New extremal Kähler metrics found on 4-manifolds with U(2) symmetry.

Analytic plane curves determine unique conformal coordinates.

A new geometric cocycle measures mass of hyperbolic manifolds.

Study on solitons in complex Riemannian manifolds with specific properties.

Study on Yamabe solitons on specific complex manifolds, focusing on torse-forming vector fields.

Study LCS structures on Lie algebras of type I, proving trivial Morse-Novikov cohomology and constructing solvmanifolds.

Topological complexity for closed 1-forms

Study on Riemannian Poisson warped product spaces and their properties.

Researchers compute Ricci curvature on noncommutative 3-tori.

This article deals with a continuous closed 1-form defined on a CW-complex. In particular, we show Lusternik-Schnirelmann type theory on continuous closed 1-forms which is related to gradient-like flows. M.Farber defined a continuous closed 1-form and a category with a respect to a cohomology class and constructed a Lu…

The paper explores parallel 1-forms on special Finsler manifolds and their properties.

New insights into cohomology of closed 1-forms.

Indices of vector fields and 1-forms studied for singular varieties and actions.

Let $(M,g)$ be a compact Riemannian manifold on which a trace-free and divergence-free $σ\in W^{1,p}$ and a positive function $τ\in W^{1,p}$, $p > n$, are fixed. In this paper, we study the vacuum Einstein constraint equations using the well known conformal method with data $σ$ and $τ$. We show that if no solution exis…

Study on $4$-dimensional almost-Hermitian manifolds, proving $\overline\partial$-harmonic forms invariant under certain metrics.

The Lee-Gauduchon cone is a convex cone of cohomology classes for complex manifolds.

Geodesic X-ray transform proves injective for smooth one-forms on gas giant manifolds.

Paper studies new Finsler metrics with specific curvature properties.

In this paper we prove a useful formula for the graded commutator of the Hodge codifferential with the left wedge multiplication by a fixed $p$-form acting on the de Rham algebra of a Riemannian manifold. Our formula generalizes a formula stated by Samuel I. Goldberg for the case of 1-forms. As first examples of applic…

Lyapunov 1-forms on orbifolds help understand flows on compact spaces.