arXiv research

Analyzes properties of Hopf manifolds from analytic and metric perspectives.

Finite-type surfaces have a topological Hopf property.

The Hopf invariant is linked to null-homotopy properties of maps.

We construct and prove a diagrammatic version of the Duflo isomorphism between the invariant subalgebra of the symmetric algebra of a Lie algebra and the center of the universal enveloping algebra. This version implies the original for metrized Lie algebras (Lie algebras with an invariant non-degenerate bilinear form).…

The paper classifies Hopf hypersurfaces in complex quadrics with commuting Jacobi operators.

Hopf manifolds can be given lcK structures, shown by constructing a family.

We prove that the Hopf vector field is a unique one among geodesic covariantly normal unit vector fields on spheres such that the submanifold generated by the field is totally geodesic in the unit tangent bundle with Sasaki metric. As application, we give a new proof of stability (instability) of the Hopf vector field …

The paper classifies and determines properties of specific hypersurfaces in complex hyperbolic quadrics.

The curvature of almost Fuchsian immersions is concave in their Hopf differentials.

The Hopf-Rinow theorem is extended to sub-Finslerian geometry.

It was shown by Jim Davis that a 2-component link with Alexander polynomial one is topologically concordant to the Hopf link. In this paper, we show that there is a 2-component link with Alexander polynomial one that has unknotted components and is not smoothly concordant to the Hopf link, answering a question of Jim D…

We introduce a Hopf algebroid associated to a proper Lie group action on a smooth manifold. We prove that the cyclic cohomology of this Hopf algebroid is equal to the de Rham cohomology of invariant differential forms. When the action is cocompact, we develop a generalized Hodge theory for the de Rham cohomology of inv…

Symmetric hypersurfaces with constant mean curvature are spheres.

This paper updates knot invariants using Hopf algebras and categorifies their structure.

The analysis of classical consensus algorithms relies on contraction properties of adjoints of Markov operators, with respect to Hilbert's projective metric or to a related family of seminorms (Hopf's oscillation or Hilbert's seminorm). We generalize these properties to abstract consensus operators over normal cones, w…

Heinz Hopf's famous fibrations of the 2n+1-sphere by great circles, the 4n+3-sphere by great 3-spheres, and the 15-sphere by great 7-spheres have a number of interesting properties. Besides providing the first examples of homotopically nontrivial maps from one sphere to another sphere of lower dimension, they all share…

In this paper, we define a certain "proportional volume property" for an unit vector field on a spherical domain in S3. We prove that the volume of these vector fields has an absolute minimum and this value is equal to the volume of the Hopf vector field. Some examples of such vector fields are given. We also study the…

Solves an old problem by showing round spheres are the only compact surfaces with specific curvature properties.

Oleg Viro studied in arXiv:math/0204290 two interpretations of the (multivariable) Alexander polynomial as a quantum link invariant: either by considering the quasi triangular Hopf algebra associated to $U_q sl(2)$ at fourth roots of unity, or by considering the super Hopf algebra $U_q gl(1|1)$. In this paper, we show …

New invariant calculates 4-manifolds using trisection diagrams and combings.

We consider diagrams of links in $S^2$ obtained by projection from $S^3$ with the Hopf map and the minimal crossing number for such diagrams. Knots admitting diagrams with at most one crossing are classified. Some properties of these knots are exhibited. In particular, we establish which of these knots are algebraic an…

Paper analyzes a new Hopf-Lax semigroup in metric spaces.

We study the compact Hermitian spin surfaces with positive conformal scalar curvature on which the first eigenvalue of the Dolbeault operator of the spin structure is the smallest possible. We prove that such a surface is either a ruled surface or a Hopf surface. We give a complete classification of the ruled surfaces …

Poincaré-Hopf theorem extended to Filippov vector fields on 2D manifolds.

Researchers describe character varieties for Hopf links, proving geometric properties.

We study the horizontal Laplacian $Δ^H$ associated to the Hopf fibration $S^3\to S^2$ with arbitrary Chern number $k$. We use representation theory to calculate the spectrum, describe the heat kernel and obtain the complete heat trace asymptotics of $Δ^H$. We express the Green functions for associated Poisson semigroup…

Singular fibrations over surfaces generalize Lefschetz fibrations and have new construction methods.

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…

We deal with Riemannian properties of the octonionic Hopf fibration S^{15}-->S^8, in terms of the structure given by its symmetry group Spin(9). In particular, we show that any vertical vector field has at least one zero, thus reproving the non-existence of S^1 subfibrations. We then discuss Spin(9)-structures from a c…

Investigates BNSR invariants of link and knot groups, proving specific properties.

Proves uniqueness of capillary disks in 3D domains.

New principles prove precompactness of domains with lower Ricci curvature bound.

Characterizes conformal classes of tori using differential geometry.

Non-linear Hopf manifolds can be embedded into linear ones and admit LCK metrics.

This paper analyzes complex equilibria in a networked bivirus epidemic model.

Analyses cohomology relations for moving frames and coframes.

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.

Algorithm calculates Hopf invariant for simplicial mappings.

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…

We define a partition of the space of projectively flat metrics in three classes according to the sign of the Chern scalar curvature; we prove that the class of negative projectively flat metrics is empty, and that the class of positive projectively flat metrics consists precisely of locally conformally flat-Kähler met…

Among all torus links, we characterise those arising as links of simple plane curve singularities by the property that their fibre surfaces admit only a finite number of cutting arcs that preserve fibredness. The same property allows a characterisation of Coxeter-Dynkin trees (i.e., $A_n$, $D_n$, $E_6$, $E_7$ and $E_8$…

Introduces intrinsic Hopf-Lax semigroup linking to intrinsic slope.

Nilpotent quandles have simple characterizations and are closely related to nilpotency.

New upper bound for non-trivial links in a cube, geometric approach.

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…

In this work we obtain some geometric properties of biconservative surfaces into a Riemannian manifold. In particular, we shall study the relationship between biconservative surfaces and the holomorphicity of a generalized Hopf function. Also, we give a complete classification of CMC biconservative surfaces in a $4$-di…

Classifies Legendrian Hopf links in lens spaces.