Non-linear Hopf manifolds can be embedded into linear ones and admit LCK metrics.
problem Understanding non-linear Hopf manifolds and their properties.
method Holomorphic embeddings and LCK metrics.
result Non-linear Hopf manifolds admit LCK metrics.
Automorphism groups of Hopf manifolds are finite and have a bounded order.
problem Understanding the structure of automorphism groups of Hopf manifolds.
method Proved that the automorphism groups of Hopf manifolds are Jordan.
result Automorphism groups of Hopf manifolds are finite and have a bounded order.
Analyzes properties of Hopf manifolds from analytic and metric perspectives.
problem Properties of Hopf manifolds
method Analytic and metric structure
result Reviews old and new properties of Hopf manifolds
Futaki invariant vanishes on Hopf manifolds.
problem Obtaining Kähler-Einstein metrics on compact manifolds.
method Generalized Futaki invariant to Hopf manifolds and proved its vanishing.
result Futaki invariant vanishes on Hopf manifolds.
We classify nonsingular holomorphic foliations of dimension and codimension one on certain Hopf manifolds. More general, we prove that all nonsingular codimension one distributions on intermediary or generic Hopf manifolds are integrable and has holomorphic integral first. Also, we prove some results about singular hol…
Hopf manifolds can be given lcK structures, shown by constructing a family.
problem Constructing locally conformally Kaehler structures for Hopf manifolds.
method Analytic family construction and application of Ornea-Verbitsky's theorem.
result Hopf manifolds can be endowed with lcK structures.
New proofs for complex Hopf manifolds using geometric structures.
problem Proving properties of complex Hopf manifolds.
method Constructing integrable holomorphic G-structures and flat holomorphic Cartan geometries.
result Provided a new proof of flat holomorphic Cartan geometries on complex Hopf manifolds.
Invariants for 4-manifolds from Hopf group-algebras.
problem Constructing invariants for flat connections on 4-manifolds.
method Using finite type involutory quasitriangular Hopf G-algebras and coloring Kirby diagrams. result Invariants defined for 4-manifolds and connections.
The Kuperberg invariant is a topological invariant of closed 3-manifolds based on finite-dimensional Hopf algebras. In this paper, we initiate the program of constructing 4-manifold invariants in the spirit of Kuperberg's 3-manifold invariant. We utilize a structure called a Hopf triplet, which consists of three Hopf a…
Involutory Hopf group-coalgebras provide new invariants for 4-manifold bundles.
problem Developing invariants for flat bundles over 4-manifolds.
method Utilizing Hopf G-triplets and colored trisection diagrams. result Involutory Hopf G-triplets yield well-defined invariants of G-colored trisection diagrams. Holomorphic vector bundles on Hopf manifolds admit flat connections.
problem Understanding flat connections on holomorphic vector bundles over Hopf manifolds.
method Defining resonant and non-resonant Mall bundles, proving the existence of flat connections on non-resonant bundles, and applying the Poincare-Dulac theorem.
result Non-resonant Hopf manifolds are linearizable, generalizing Kodaira's result.
Hopf's theorem generalized to curved spaces.
problem Extending Hopf's theorem to non-Euclidean spaces.
method Curvature flows and warped product manifolds.
result Generalized Hopf's theorem to specific curved spaces.
Invariants of 3-manifolds using modified Hopf G-coalgebra.
problem Constructing invariants for 3-manifolds.
method Purely Hopf G-coalgebra construction with modified integral.
result New invariants for 3-manifolds.
Generalizes Hopf degree theorem to nontrivial bundles.
problem Classifying maps from manifolds to spheres.
method Generalization of Hopf degree theorem to nontrivial bundles.
result Classifies sections of nontrivial n-sphere bundles. In a three-dimensional Riemannian manifold M that admits a unit Killing vector field ξ, we regard ξ as a magnetic vector field. A magnetic Hopf surface is a surface obtained by Lie dragging the magnetic curve with ξ. Then we characterize Sasakian structure on M from magnetic Hopf surfaces. That is, we show that i…
Every holomorphic effective parabolic or reductive geometry on a domain over a Stein manifold extends uniquely to the envelope of holomorphy of the domain. This result completes the open problems of my earlier paper on extension of holomorphic geometric structures on complex manifolds. We use this result to classify th…
New gauge invariants from framed 3-manifolds match Hopf algebra indicators.
problem Computing indicators for Hopf algebras.
method Using Kuperberg invariants from framed 3-manifolds.
result Kuperberg invariants match higher Frobenius-Schur indicators of Hopf algebras.
Paper connects two invariants of 3D manifolds using Hopf algebras.
problem Establishing a relation between two invariants of 3D manifolds.
method Using spherical Hopf algebras and their Drinfeld doubles, the paper connects the chromatic spherical invariant and the Hennings-Kauffman-Radford invariant.
result The chromatic spherical invariant is equal to the Hennings-Kauffman-Radford invariant for a specific type of Hopf algebra.
Proves Poincaré duality for Hopf algebroids with bijective antipode.
problem Proving Poincaré duality for Hopf algebroids.
method Using twisted Poincaré duality and bijective antipode properties.
result Recovering and extending known Poincaré dualities for Hopf algebroids.
Extends Hopf-Rinow theorem to semi-Riemannian spacetimes.
problem Generalizing Hopf-Rinow theorem to compact Lorentzian manifolds.
method Develops null distance for proper cone structures and (n−ν,ν)-spacetimes. result Generalizes Hopf-Rinow theorem to a new class of semi-Riemannian manifolds.
Given a group G, we use involutary Hopf G-coalgebras to define a scalar invariant of flat G-bundles over 3-manifolds. When G=1, this invariant equals to the one of 3-manifolds constructed by Kuperberg from involutary Hopf algebras. We give examples which show that this invariant is not trivial.
In the 90s, based on presentations of 3-manifolds by Heegaard diagrams, Kuperberg associated a scalar invariant of 3-manifolds to each finite dimensional involutory Hopf algebra over a field. We generalize this construction to the case of involutory Hopf algebras in arbitrary symmetric monoidal categories admitting cer…
Researchers decompose curvature to confirm Hopf conjecture and prove new rigidity theorems.
problem Confirming the Hopf conjecture on compact Riemannian manifolds of even dimension.
method Decomposing the curvature operator into Hermitian components and developing eigenvalue criteria for sectional curvature.
result Prove vanishing theorems for Betti numbers under integral bounds on the Weyl tensor and confirm the Hopf conjecture for manifolds with sufficiently small Weyl curvature.
Study the embedding space of a Hopf link in 3D and 3-manifolds.
problem Homotopy equivalence of embedding spaces of Hopf links.
method Prove homotopy equivalence using subspace inclusions and manifold properties.
result Embedding spaces are homotopy equivalent to specific 3-manifolds.
New quantum invariant for framed 3-manifolds using ideal triangulations.
problem Quantum invariants of framed 3-manifolds with vanishing first Betti number.
method Based on ideal triangulations and Hopf algebras, using the pentagon equation and graphical representations.
result Construction of a new quantum invariant for closed framed 3-manifolds.
The Kuperberg invariant is shown to be gauge invariant for certain framed 3-manifolds.
problem Exploring gauge invariance of the Kuperberg invariant for specific 3-manifolds.
method Using hyperbolic 3-manifolds and finite-dimensional Hopf algebras.
result First examples of gauge invariants of general finite-dimensional Hopf algebras via topological methods.
In this note we study logarithmic transformations in the sense of differential topology on two fibers of the Hopf surface. It is known that such transformations are susceptible to yield exotic smooth structures on four-manifolds. We will show here that this is not the case for the Hopf surface, all integer homology Hop…
We investigate isometric immersions of locally conformally Kaehler metrics into Hopf manifolds. In particular, we study Hopf-induced metrics on compact complex surfaces.
New invariant calculates 4-manifolds using trisection diagrams and combings.
problem Calculating non-semisimple 4-manifold invariants.
method Using trisection diagrams and combings of the trisection surface.
result Invariant calculated for Stein nuclei, generalizing earlier semisimple version.
The paper studies bifurcations in discrete dynamical systems on manifolds.
problem Understanding bifurcations in discrete dynamical systems on manifolds.
method Topological techniques based on concentricity of manifolds.
result General result for attractors in n-dimensional manifolds.
This paper updates knot invariants using Hopf algebras and categorifies their structure.
problem Defining and understanding quantum invariants of knots and three-manifolds.
method Abstract description of categorical structures involving Hopf algebras and their centers.
result The Hopf algebraic center of a knot's image is central to the invariant.
Higher order higher spin operators are generalizations of kth-powers of the Dirac operator. In this paper, we study higher order higher spin operators defined on some conformally flat manifolds, namely cylinders and Hopf manifolds. We will also construct the kernels of these operators on these manifolds.
The paper computes lens spaces resulting from rational surgeries on Hopf links.
problem Understanding rational surgeries on Hopf links in 3-sphere.
method Calculus of continued fractions.
result Explicit computation of resulting lens spaces.
An LCK manifold with potential is a compact quotient M of a Kahler manifold X equipped with a positive plurisubharmonic function f, such that the monodromy group acts on X by holomorphic homotheties and maps f to a function proportional to f. It is known that M admits an LCK potential if and only if it can be holomor…
Extends Tian theorem to Vaisman manifolds for approximations.
problem Approximating Vaisman metrics by immersions/embeddings.
method Study Vaisman metrics on compact manifolds.
result Extend Tian's theorem to Vaisman manifolds.
Extends Hopf's theorem to de Sitter-Schwarzschild and Reissner-Nordstrom manifolds.
problem Finding constant mean curvature surfaces in specific spacetimes.
method Partial differential equations in the complex plane, generalizing holomorphy.
result Extends Hopf's theorem to new spacetime geometries.
Holomorphic structures on quantum flag manifolds uniquely defined.
problem Defining unique holomorphic structures on quantum flag manifolds.
method Constructing covariant q-deformed holomorphic structures. result Holomorphic structures are unique for simple relative Hopf modules.
We state and prove a generalization of the Poincaré-Hopf index theorem for manifolds with boundary. We then apply this result to non-vanishing complex vector fields.
Crane and Frenkel proposed a state sum invariant for triangulated 4-manifolds.They defined and used new algebraic structures called Hopf categories for their construction. Crane and Yetter studied Hopf categories and gave some examples using group cocycles that are associated to the Drinfeld double of a finite group. I…
Hennings and Kuperberg defined quantum invariants ZHenn and ZKup for closed oriented 3-manifolds based on certain Hopf algebras, respectively. When the Hopf algebras are semisimple, it is shown that ZKup=∣ZHenn∣2. In this paper, we present a new proof of this equality.
We prove a 20-year-old conjecture concerning two quantum invariants of three manifolds that are constructed from finite dimensional Hopf algebras, namely, the Kuperberg invariant and the Hennings-Kauffman-Radford invariant. The two invariants can be viewed as a non-semisimple generalization of the Turaev-Viro-Barrett-W…
The general method of Reshetikhin and Turaev is followed to develop topological invariants of closed, connected, orientable 3-manifolds from a new class of algebras called pseudo-modular Hopf algebras. Pseudo-modular Hopf algebras are a class of Z_2-graded ribbon Hopf algebras that generalise the concept of a modular H…
Holonomy group of Bismut connection on Vaisman manifolds is studied.
problem Analyzing the holonomy group of Bismut connection on Vaisman manifolds.
method Proved and computed the holonomy group for Vaisman manifolds, including solvmanifolds and Hopf manifolds.
result Holonomy group of Bismut connection on Vaisman manifolds is contained in U(n-1).
New examples of real hypersurfaces found in complex hyperbolic quadrics.
problem Existence of specific types of real hypersurfaces in complex hyperbolic quadrics.
method Construction of a one-parameter family of homogeneous Hopf hypersurfaces.
result First known examples of real hypersurfaces with integrable maximal complex subbundle in irreducible Kahler manifolds.
We investigate slice-quaternionic Hopf surfaces. In particular, we construct new structures of slice-quaternionic manifold on S1×S7, we study their group of automorphisms and their deformations.
Poincaré-Hopf theorem extended to Filippov vector fields on 2D manifolds.
problem Extending Poincaré-Hopf theorem to Filippov vector fields.
method Introducing new index definition for Filippov vector fields, including singularities.
result Established a variant of Hairy Ball Theorem for Filippov vector fields.
Proves a 1930s Hopf conjecture about positive curvature manifolds.
problem Even-dimensional compact Riemannian manifolds with positive sectional curvature and high isometry rank.
method Reduces to a representation theoretic problem involving torus representations.
result Proves the Hopf conjecture under specific conditions.
The Hopf-Rinow theorem is extended to sub-Finslerian geometry.
problem Extending the Hopf-Rinow theorem to sub-Finslerian manifolds.
method Investigation of sub-Finslerian bundle, exponential map, and Legendre transformation.
result Established a relation between completeness, geodesic completeness, and compactness in sub-Finslerian geometry.