The paper studies canonical 1-forms on Lie-Rinehart structures of Weil bundles.
problem Properties of differential operators on Weil bundles.
method Construction of canonical 1-forms for Lie-Rinehart structures on Weil bundles.
result Construction of canonical 1-forms associated with Lie-Rinehart structures on Weil bundles.
Constructs deformations of Vaisman manifolds preserving foliations.
problem Deforming Vaisman manifolds while maintaining their canonical foliations.
method Uses a basic 1-form with specific properties to construct transverse deformations.
result Basic 1-forms exist in abundance for constructing deformations.
Study complex solvmanifolds with trivial canonical bundle and hypercomplex geometry.
problem Characterize and produce examples of complex solvmanifolds with trivial canonical bundle.
method Characterize invariant trivializing sections using Koszul 1-form, provide algebraic obstructions, and exhibit specific examples.
result New examples of complex solvmanifolds with trivial canonical bundle and algebraic obstructions for triviality.
On 5-dimensional almost contact B-metric manifolds, the form of any Kähler-type tensor (i.e. a tensor satisfying the properties of the curvature tensor of the Levi-Civita connection in the special class of the parallel structures on the manifold) is determined. The associated 1-forms are derived by the scalar curvature…
Study canonical deformations of complex forms and their cohomology properties.
problem Understanding canonical deformations and cohomology of complex manifolds.
method Analyzes canonical Aeppli deformations and their relations to deformed cohomology.
result Proves the jumping formula for deformed Aeppli cohomology and constant dimension conditions.
Study Fano fibrations and Kähler-Einstein metrics on their bases.
problem Understand geometric structures and metrics on Fano fibrations.
method Construct (1,1)-forms and solve twisted Kähler-Einstein equations. result Singular Kähler metrics on Fano fibrations satisfy twisted Kähler-Einstein equations.
Researchers compute the spectrum of Hodge-Laplacian on 1-forms for SU(2) and SO(3).
problem Computing the spectrum of the Hodge-Laplacian on 1-forms for homogeneous 3-spheres.
method Explicit computation of eigenvalues for Berger 3-spheres and general homogeneous metrics on SU(2) and SO(3).
result The spectrum on 1-forms determines the metric up to isometry.
Topology of Foliations of the Riemann Surfaces given by the real part of generic holomorphic 1-forms, is studied. Our approach is based on the notion of Transversal Canonical Basis of Cycles (TCB) instead of using just one closed transversal curve as in the classical approach of the ergodic theory. In some cases the TC…
New formula found for a unique invariant 8-form on Riemannian manifolds with Spin(9) structure.
problem Finding a new explicit algebraic formula for a unique invariant 8-form.
method Generalizing the standard Kähler 2-form expression, constructing the invariant 8-form from octonion-valued coordinate 1-forms.
result A new explicit algebraic formula for the Spin(9)-invariant 8-form. Study heat flow inequalities on 1-forms in RCD spaces.
problem Heat flow inequalities on 1-forms in RCD spaces.
method Analyzes heat flow (Ht) and its properties on cotangent modules over RCD spaces. result Establishes various Lp-properties and spectrum inclusions for the heat flow. Study on metrizability and Ricci-flatness of Finsler spaces with Kropina metrics.
problem Investigating metrizability and Ricci-flatness of Finsler spaces with specific metrics.
method Analyzing the Levi-Civita connection, covariant derivative of 1-forms, and using cohomology groups.
result Explicitly obtained all Ricci-flat, locally metrizable m-Kropina metrics in (3+1)D.
We solve higher-order morphisms for twisted Courant algebras.
problem Construct canonical L∞-morphisms for higher Courant algebroids. method Develop a general framework for arbitrary r. result Affirmative answer to Zambon's question for higher degrees.
Topological complexity for closed 1-forms
problem Topological complexity for closed 1-forms
method Introduce and study a corresponding version of topological complexity
result Establish analogues of basic properties of ordinary topological complexity
New extremal Kähler metrics found on 4-manifolds with U(2) symmetry.
problem Finding new extremal Kähler metrics on 4-manifolds with U(2) symmetry.
method Analyzing U(2)-invariant metrics, showing they are conformal to two separate Kähler metrics, leading to ambiKähler structures. result New complete extremal Kähler metrics found on specific 4-manifolds.
Article studies continuous closed 1-forms and their zero points on CW-complexes.
problem Understanding zero points of continuous closed 1-forms on topological spaces.
method Defines continuous closed 1-forms, constructs a category, and applies dynamical systems.
result Relates the number of zero points to the category with respect to a cohomology class.
Study on Riemannian Poisson warped product spaces and their properties.
problem Characterizing and understanding Riemannian Poisson warped product spaces.
method Formal treatment of Killing and 2-Killing 1-forms on Riemannian Poisson manifolds, including Bochner type results.
result Characterization of 2-Killing 1-form on (R2,g,Π) and Bochner type results on compact spaces. The paper explores parallel 1-forms on special Finsler manifolds and their properties.
problem Investigating parallel 1-forms on specific Finsler manifolds.
method Analyzing Landsberg manifolds, metrizability freedom, and specific Finsler metrics.
result Landsberg surfaces with parallel 1-forms are necessarily Berwaldian, and the metrizability freedom is at least 2.
Let (Mn,g) be a closed, connected, oriented, C∞, Riemannian, n-manifold with a transversely oriented foliation $\boldkey F$. We show that if {X,Y} are basic vector fields, the leaf component of [X,Y], $\Cal{V}[X,Y]$, has vanishing leaf divergence whenever $κ\wedge χ_{\boldkey F}$ is a c…
New insights into cohomology of closed 1-forms.
problem Understanding twisted cohomology of closed 1-forms.
method Construction of examples and analysis of fundamental group representations.
result Non-trivial twisted cohomology of nowhere-vanishing 1-forms.
Indices of vector fields and 1-forms studied for singular varieties and actions.
problem Understanding indices of vector fields and 1-forms in various contexts.
method Generalization to singular varieties and actions of finite groups.
result New insights into indices of vector fields and 1-forms.
Lyapunov 1-forms on orbifolds help understand flows on compact spaces.
problem Understanding flows on orbifolds using Lyapunov 1-forms.
method Introducing Lyapunov 1-forms, using asymptotic cycles and chain-recurrent sets.
result Existence of a Lyapunov 1-form in a prescribed cohomology class for compact orbifolds.
If a closed 3-manifold M supports a closed, nonsingular, irrational 1-form which linearly deforms into contact forms, then M supports a K-contact form. On the 3-torus, a closed nonsingular 1-form deforms linearly into contact forms if and only if it is a fibration 1-form. on any other 2-torus bundle over the circle, ev…
In the spirit of noncommutative geometry we construct all inequivalent vector bundles over the (2,2)-dimensional supersphere S2,2 by means of global projectors p via equivariant maps. Each projector determines the projective module of finite type of sections of the corresponding `rank 1' supervector bundle ove…
New metrics produce discrete zero sets for nondegenerate harmonic forms.
problem Creating metrics to produce discrete zero sets for nondegenerate harmonic forms.
method Metric perturbation to produce new nondegenerate harmonic forms with discrete zero sets.
result Existence of metrics producing discrete zero sets for nondegenerate harmonic forms.
S.P.Novikov developed an analog of the Morse theory for closed 1-forms. In this paper I suggest an analog of the Lusternik - Schnirelman theory for closed 1-forms.
Generalizes reductive homogeneous spaces to arbitrary Lie groups using gauge theory.
problem Classify admissible triples (PoπM,α,A) on principal bundles. method Gauge-theoretical approach, differential system, integrability condition.
result Classifies triples associated with real forms of complex Lie groups.
The study resolves a conjecture about harmonic forms on compact manifolds.
problem Finding non-degenerate Z2-harmonic 1-forms on compact manifolds. method Develops a gluing theorem for non-degenerate Z2-harmonic 1-forms on compact manifolds. result Proves the existence of non-degenerate Z2-harmonic 1-forms on compact manifolds with positive first Betti number. The paper defines vector 1-forms on Finsler manifolds and constructs connections.
problem Characterizing conservative connections on Finsler manifolds.
method Defining conservative semibasic vector 1-forms and constructing connections.
result A correspondence between torsion-free semibasic vector 1-forms and vertical vector fields.
Calabi surgery modifies Z/2 harmonic 1-forms using 2-valued 1-forms.
problem Modifying Z/2 harmonic 1-forms under weak regularity assumptions.
method Calabi surgery method, involving cutting and pasting closed 2-valued 1-forms.
result Flexible construction and modification of Z/2 harmonic 1-forms.
The paper proves vanishing and finiteness theorems for p-harmonic 1-forms.
problem The vanishing and finiteness of p-harmonic 1-forms on submanifolds.
method Using BiRic curvature conditions to prove theorems.
result Theorems on vanishing and finiteness of p-harmonic 1-forms.
Study topological properties of foliations induced by closed 1-forms on orbifolds.
problem Characterize the topology of foliation leaves induced by closed 1-forms on orbifolds.
method Establish criteria for the compactness of foliation leaves and extend a topological result to orbifolds.
result Criteria for the compactness and coexistence of foliation leaves are established.
The study characterizes a complex curve of residueless meromorphic differentials on elliptic curves.
problem Characterizing the locus of residueless meromorphic differentials on elliptic curves.
method Multi-scale compactification of strata, formulas for genus and degree of maps, distinguishing components.
result Complete classification of connected components of residueless loci in exceptional strata.
The study finds nondegenerate harmonic 1-forms using symmetry conditions.
problem Existence of nondegenerate harmonic 1-forms over Riemannian manifolds.
method Utilizing Z3 symmetry to establish topological conditions. result Found nondegenerate Z2 harmonic 1-forms over branched coverings of links. We discuss controlled connectivity properties of closed 1-forms and their cohomology classes and relate them to the simple homotopy type of the Novikov complex. The degree of controlled connectivity of a closed 1-form depends only on positive multiples of its cohomology class and is related to the Bieri-Neumann-Strebel…
We propose an extension of the structure equation for constant mean curvature (CMC) surfaces in a three dimensional Riemannian space form to the associated CMC hierarchy of evolution equations by the higher-order commuting symmetries. Via the canonical formal Killing field, considered as an infinitely prolonged and loo…
Study Euler obstruction of 1-forms on determinantal singularities.
problem Understanding the Euler obstruction of 1-forms on determinantal singularities.
method Investigation of connections between local Euler obstruction and PHN index.
result Explicit computations of Euler obstruction for specific singularities.
Vanishing theorem for L2-harmonic forms on Riemannian manifolds with parallel 1-form.
problem Proving vanishing of L2-harmonic forms on Riemannian manifolds with a parallel 1-form. method Using L2 Morse-Novikov cohomology and a vanishing theorem. result The L2-harmonic forms on the manifold are identically zero. The paper studies local normal forms of singular contact forms and primitive 1-forms.
problem Local normal forms of singular contact forms and primitive 1-forms.
method Combines classical normalization techniques and toric approach.
result Extends and improves previous results on first-order contact forms and primitive 1-forms.
We use the Frölicher-Nijenhuis formalism to reformulate the inverse problem of the calculus of variations for a system of differential equations of order 2k in terms of a semi-basic 1-form of order k. Within this general context, we use the homogeneity proposed by Crampin and Saunders in [14] to formulate and discuss t…
Study existence of harmonic 1-forms on Calabi-Yau manifolds.
problem Tackles existence of harmonic 1-forms on Calabi-Yau manifolds.
method Uses neural networks to approximate metrics and harmonic 1-forms.
result Suggests existence of harmonic 1-forms on some Calabi-Yau manifolds.
The study examines hyperbolic 3-manifolds with uniform spectral gaps for coclosed 1-forms.
problem Understanding the spectral gap for coclosed 1-forms in hyperbolic 3-manifolds.
method Constructing sequences of manifolds and analyzing their spectral properties and homology growth.
result Sequences of hyperbolic manifolds can have uniform spectral gaps for coclosed 1-forms but unbounded torsion homology growth.
The paper constructs harmonic 1-forms on 3-manifolds with cylindrical necks.
problem Stabilizing Z/2-harmonic 1-forms on closed 3-manifolds.
method Explicit construction of harmonic 1-forms by modifying metrics near links.
result Construction of harmonic 1-forms degenerating to manifolds with cylindrical ends.
We prove in this article that given a linearly concave domain D in the projective space CPn, a 1-dimensional comlex analytic set V in D, and a meromorphic 1-form φ on V, V is a subset of an algebraic variety of CPn and φ is the restriction to V of an algebraic 1-form on $\Bbb{CP}^{…
Some years ago Moshé Flato pointed up that it could be interesting to develop the Nambu's idea to generalize Hamiltonian mechanic. An interesting new formalism in that direction was proposed by T. Takhtajan. His theory gave new perspectives concerning deformation quantization, and many authors have developed its mathem…
The paper suggests new topological lower bounds for the number of zeros of closed 1-forms within a given cohomology class. The main new technical tool is the deformation complex, which allows to pass to a singular limit and reduce the original problem with a closed 1-form to a traditional problem with a Morse function.…
The paper classifies (α,β)-metrics with vanishing Douglas curvature.
problem Classifying (α,β)-metrics with vanishing Douglas curvature.
method Using β-deformations to classify metrics with vanishing Douglas curvature.
result Conformal 1-forms of Riemannian metrics play a key role, and an effective way to construct such 1-forms is provided by β-deformations.
The paper studies almost complex structures on ACH Einstein manifolds and finds deformation results.
problem The study of canonical almost complex structures on ACH Einstein manifolds.
method Variational problem with the Dolbeault Laplacian acting on (0,1)-forms. result Deformation results of Einstein ACH metrics associated with critical almost complex structures.
The study classifies rational 1-forms on the Riemann sphere with simple poles.
problem Classifying rational 1-forms on the Riemann sphere with specified pole conditions.
method Recognized three equivalent atlases, proved submanifold properties, and used PSL(2,C) action.
result Quotients of isochronous 1-forms admit stratified orbit types.