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.
Analyses cohomology relations for moving frames and coframes.
problem Relating Hopf cyclic cohomology of moving frames and coframes.
method Uses van Est analogy for DG Hopf algebras.
result Establishes cohomology isomorphism for DG Hopf algebras.
We associate to each infinite primitive Lie pseudogroup a Hopf algebra of `transverse symmetries', by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a `quantum group' counterpart of the infinite-dimensional primitive Lie algeb…
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.
Algorithm calculates Hopf invariant for simplicial mappings.
problem Computing Hopf invariant for simplicial mappings.
method Proposed an algorithm based on Whitehead's integral formula.
result Algorithm successfully computes Hopf invariant.
We refine the cyclic cohomological apparatus for computing the Hopf cyclic cohomology of the Hopf algebras associated to infinite primitive Cartan-Lie pseudogroups, and for the transfer of their characteristic classes to foliations. The main novel feature is the precise identification as a Hopf cyclic complex of the im…
Introduces intrinsic Hopf-Lax semigroup linking to intrinsic slope.
problem Understanding intrinsic Hopf-Lax semigroup and its relation to intrinsic slope.
method Introduces and proves the link between intrinsic Hopf-Lax semigroup and intrinsic slope.
result Intrinsic Hopf-Lax semigroup is a subsolution of Hamilton-Jacobi type equality.
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
We present a geometric approach, in the spirit of the Chern-Weil theory, for constructing cocycles representing the classes of the Hopf cyclic cohomology of the Hopf algebra H(n) relative to GL(n, R). This provides an explicit description of the universal Hopf cyclic Chern classes, which complements our earlier geometr…
Classifies Legendrian Hopf links in lens spaces.
problem Classifying Legendrian Hopf links in lens spaces.
method Classification up to coarse equivalence.
result Legendrian Hopf links in L(p,1) are classified.
The Hopf invariant is linked to null-homotopy properties of maps.
problem Understanding the relationship between the Hopf invariant and null-homotopy of maps.
method Using the generalized Hopf invariant and constructing smooth null-homotopies, the paper explores the relationship between the Hopf invariant and null-homotopy properties of maps.
result Sharp results on the relationship between the Hopf invariant and null-homotopy properties of maps, showing the necessity and sufficiency of certain conditions.
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 construct a Hopf action, with an invariant trace, of a bicrossed product Hopf algebra $\cH=\big( \cU(\Fg_1) \acr \cR(G_2) \big)^{\cop}$ constructed from a matched pair of Lie groups G1 and G2, on a convolution algebra $\cA=C_c^{\ify}(G_1)\rtimes G_2^δ$. We give an explicit way to construct Hopf cyclic cohomolo…
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. 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…
The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.
problem Extending the Poincaré-Hopf theorem to varieties with isolated singularities.
method Using generalizations of the Poincaré-Hopf index.
result A Poincaré-Hopf type theorem for projective varieties with isolated singularities.
For a positive Hopf plumbed arborescent Seifert surface S, we study the set of Hopf bands H⊂S, up to homology and up to the action of the monodromy. The classification of Seifert surfaces for which this set is finite is closely related to the classification of finite Coxeter groups.
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. Classifies totally geodesic submanifolds in Hopf-Berger spheres.
problem Classifying totally geodesic submanifolds in Hopf-Berger spheres.
method Investigates Hopf-Berger spheres, a special family of homogeneous spaces diffeomorphic to spheres constructed via Hopf fibrations.
result Discovered intriguing examples of totally geodesic submanifolds, including isometric submanifolds of real projective spaces and non-congruent submanifolds.
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.
New algebraic structure derived from Hopf algebra and Drinfel'd twist.
problem Developing a new algebraic structure from existing mathematical concepts.
method Extending L∞-algebra to a Hopf algebra, twisting with Drinfel'd twist, and identifying Hopf morphisms and braided morphisms. result Braided L∞-algebra is derived from the process. 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.
The paper classifies Hopf hypersurfaces with constant curvatures on complex quadrics.
problem Classifying Hopf hypersurfaces with specific curvature properties.
method Analyzing hypersurfaces on complex quadrics with at most five distinct constant principal curvatures.
result All classified hypersurfaces are open parts of homogeneous examples.
Paper proves unbounded stabilization heights of fiber surfaces using Hopf invariant.
problem Unbounded stabilization heights of fiber surfaces.
method Alternative proof using Hopf invariant.
result Proves unbounded stabilization heights of fiber surfaces.
The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.
problem Extending the Poincaré-Hopf theorem to projective varieties with isolated singularities.
method Using generalized Poincaré-Hopf indices for a projective variety with isolated determinantal singularities.
result A Poincaré-Hopf type theorem is proven for projective varieties with isolated singularities.
New framework models high-Hopf-index hopfions using generalized fold maps.
problem Modeling and understanding high-Hopf-index hopfions.
method Proposes a new mathematical framework based on generalized Hopf maps and fold maps.
result Establishes a robust bridge between singular maps and experimentally observed hopfion topologies.
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.
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…
Analogous exponential map defined for Hopf algebras.
problem Defining an exponential map for Hopf algebras.
method Analogy with Lie groups, interpretation as states, Hilbert C* bimodules, dual Hopf algebra elements.
result Multiple interpretations of exponential map values in Hopf algebras.
Study links Hopf differentials to curvature line flows on time-like CMC surfaces.
problem Understanding the relationship between Hopf differentials and curvature line flows on time-like CMC surfaces.
method Investigation of Hopf differentials and curvature line flows on time-like CMC surfaces in Lorentzian 3-space forms.
result The index of a curvature line flow at an umbilic point depends on the remainder of the Hopf differential's order modulo four.
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…
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.
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.
Study representation varieties of twisted Hopf links using combinatorial and Hodge theory.
problem Representation theory of Hopf link complements with n twists.
method Combinatorial problem and equivariant Hodge theory.
result Close formulas for E-polynomials of representation and character varieties for ranks 2 and 3.
Classifies exceptional Legendrian realizations of Hopf link connected sums.
problem Classifying exceptional Legendrian realizations of Hopf link connected sums.
method Complete coarse classification using Legendrian knot theory.
result First classification result about exceptional Legendrian representatives for Hopf link connected sums.
Paper constructs Hopf real hypersurfaces in complex hyperbolic space.
problem Existence of Hopf real hypersurfaces in complex hyperbolic space.
method Twistor spaces and para-quaternionic Kähler structure.
result Unified construction of Hopf real hypersurfaces in complex hyperbolic space.
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.
A Poincaré-Hopf Theorem for line fields with point singularities on orientable surfaces can be found Hopf's 1956 Lecture Notes on Differential Geometry. In 1955 Markus presented such a theorem in all dimensions, but Markus' statement only holds in even dimensions 2k≥4. In 1984 Jänich presented a Poincaré-Hopf th…
Study continuity equation on Hopf and Inoue surfaces, proving estimates and convergence.
problem Analyzing the continuity equation on specific complex surfaces.
method Extended La Nave-Tian's continuity equation to Hermitian setting, proving estimates and Gromov-Hausdorff convergence.
result Proved a priori estimates for solutions on Hopf and Inoue surfaces, and convergence of Inoue surfaces to a circle.
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.
The paper classifies and determines properties of specific hypersurfaces in complex hyperbolic quadrics.
problem Analyzing Hopf hypersurfaces with constant principal curvatures in complex hyperbolic quadrics.
method Classification and determination of principal curvatures for hypersurfaces with different numbers of distinct curvatures.
result Classification and determination of principal curvatures for Hopf hypersurfaces with up to four distinct values.
New tool: relative Hopf invariant for Poincaré surgery.
problem Non-simply connected Poincaré surgery.
method Relative Hopf invariant in equivariant setting.
result Established Poincaré embedding results in relative setting.
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.
The Hopf fibration is rigid among minimal maps between spheres.
problem Characterizing minimal submersions between spheres.
method Analyzing the properties of the Hopf fibration and minimal maps.
result The Hopf fibration is the only minimal submersion from S3 to S2 under certain conditions. Using the methods of moving frames and exterior differential systems, we show that there exist Hopf hypersurfaces in complex hyperbolic space CH^2 with any specified value of the Hopf principal curvature less than or equal to the corresponding value for the horosphere. We give a construction for all such hypersurfaces …
Researchers propose a new approach to the Hopf problem for aspherical varieties.
problem The Hopf problem for aspherical smooth projective varieties.
method An approach recently proposed by Liu, Maxim, and Wang in [LMW21].
result An intriguing conjecture regarding the geography of aspherical surfaces of general type.
Currents on Lie groups form a Hopf algebra structure.
problem Understanding algebraic structure of currents on Lie groups.
method Defined Hopf algebra structure on currents using convolution and wedge product.
result Explicit formulas for Hopf algebra operations on currents are derived.
It is known that a tube over a Kahler submanifold in a complex form is a Hopf hypersurface. In some sense the reverse statement is true: a connected compact generic immersed C^(2n-1) regular Hopf hypersurface in the complex projective plane is a tube iver an irreducible algebraic variety. In the complex hyperbolic spac…