Study finds unique instanton for small α values in SU(2) Hopf fibration.
problem Identifying Yang-Mills connections in the context of α-energy.
method Approximations with Yang-Mills α-energy, focusing on SU(2) Hopf fibration.
result SO(4) invariant ADHM instanton is the unique α-critical point for small α values.
A class of examples of Riemannian metrics with holonomy G_2 on compact 7-manifolds was constructed by the author in arXiv:math.DG/0012189 and later in a joint work with N.-H. Lee in arXiv:0810.0957, using a certain `generalized connected sum' of two asymptotically cylindrical manifolds with holonomy SU(3). We consider,…
Categorifies Hopf fibration using category theory.
problem Describing Hopf fibration using category theory.
method Using small categories to classify spaces and describe the Hopf map.
result Realizes Hopf map as a functor between classifying spaces.
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. 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…
Singular fibrations over surfaces generalize Lefschetz fibrations and have new construction methods.
problem Understanding and constructing singular fibrations over surfaces.
method Explains how to construct examples of singular fibrations with a single singularity and outlines previous results.
result Closed orientable 4-manifolds with large first Betti number and vanishing second Betti number do not admit singular fibrations.
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.
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…
New proof of G2 fibration over S6 using transition functions.
problem Proving the fibration corresponds to a generator of π5(mSU(3)). method Explicit formulas and transition functions for mSU(3) bundle G2oS6. result Obtained a new proof of the fibration's generator.
Link groups can only have certain SU(2) representations.
problem Characterizing SU(2) representations of link groups.
method Analyzing the structure of link groups and their representations in SU(2).
result Irreducible SU(2) representations of link groups are restricted to specific configurations.
We compute the SU(2) Casson-Lin invariant for the Hopf link and determine the sign in the formula of Harper and Saveliev relating this invariant to the linking number.
We investigate the deformation theory of a class of generalized calibrations in Riemannian manifolds for which the tangent bundle has reduced structure group U(n), SU(n), G_2 and Spin(7). For this we use the property of the associated calibration form to be parallel with respect to a metric connection which may have no…
A fibration of a Riemannian manifold is fiberwise homogeneous if there are isometries of the manifold onto itself, taking any given fiber to any other one, and preserving fibers. Examples are fibrations of Euclidean n-space by parallel n-planes, and the Hopf fibrations of the round n-sphere by great n-spheres. In this …
A new liquid crystalline texture is proposed using gnomonic projection of the Hopf fibration.
problem Creating bend-free textures in flat space from 3-sphere Hopf fibration.
method Geodesic-preserving gnomonic projection of the Hopf fibration.
result A new liquid crystalline phase with only splay and twist.
Classifies surfaces of section for Seifert fibrations.
problem Classifying surfaces of section for Seifert fibrations.
method Discussing branched coverings and relating surfaces of section to algebraic curves.
result Relates surfaces of section to algebraic curves in weighted complex projective planes.
The main goal of this work is to study the sub-Laplacian of the unit sphere which is obtained by lifting with respect to the Hopf fibration the Laplacian of the quaternionic projective space. We obtain in particular explicit formulas for its heat kernel and deduce an expression for the Green function of the conformal s…
Researchers describe character varieties for Hopf links, proving geometric properties.
problem Character variety geometry of Hopf links with n twists. method Geometric descriptions of irreducible and totally reducible representations.
result Complete geometric description of SU(2)-character variety for r=2. The rotation angle of a rolling disc is shown to be a geometric phase related to the Hopf fibration.
problem Understanding the geometric nature of rotation angles in kinematic models.
method Using the Hopf fibration and Gauss map, the geometric phase is decomposed into dynamical and geometric components.
result The geometric phase of rotation is described as the holonomy of the Hopf fibration.
In this paper, we describe a new surprising example of a fibration of the Clifford torus S3 x S3 in the 7-sphere by great 3-spheres, which is fiberwise homogeneous but whose fibers are not parallel to one another. In particular it is not part of a Hopf fibration. A fibration is fiberwise homogeneous when for any two fi…
Supports conjecture about harmonic maps from S³ to S².
problem Tackles conjecture about harmonic maps from S³ to S².
method Uses conditions on Hessian and singular values to prove validity.
result Obtains pinching theorem for minimal hypersurfaces in the sphere.
Study subelliptic heat kernel on lifted sphere from octonionic projective space.
problem Analyzing sub-Laplacian on lifted sphere from octonionic projective space.
method Explicit formulas for heat kernel and Green function derived.
result Explicit formulas for heat kernel and Green function.
The paper finds new Ricci solitons from Hopf fibrations.
problem Finding new steady and expanding Ricci solitons.
method Analyzing cohomogeneity one Ricci solitons with specific isotropy representations.
result Existence of parameter families of non-homothetic complete steady and expanding Ricci solitons.
Symmetric minimal surfaces in spheres are constructed using symmetries of the Hopf fibration.
problem Constructing symmetric minimal surfaces in 3D spheres.
method Using Lawson's theorem and symmetries of the Hopf fibration.
result New minimal surfaces with genera 9, 25, 49, 121, 121, 361, and 841 are constructed.
Study shows crossing numbers for algebraic knots can differ by arbitrarily large amounts.
problem Comparing two crossing number definitions for algebraic knots.
method Analyzed Hopf fibration and complex singularities to compare crossing numbers.
result Difference between crossing numbers can be arbitrarily large.
Here we study geodesics connecting two given points on odd-dimensional spheres respecting the Hopf fibration. This geodesic boundary value problem is completely solved in the case of 3-dimensional sphere and some partial results are obtained in the general case. The Carnot-Carathéodory distance is calculated. We also p…
Study on flag manifolds using Hopf fibration to find geometrically special 2-spheres.
problem Computing and characterizing 2-spheres in flag manifolds.
method Hopf fibration, invariant geometry, Weyl group action.
result Generators of second homotopy group have invariant geometry and are classified.
Synthetic construction of Hopf fibration in 4D space.
problem Visualizing 4D objects in 3D space.
method Double orthogonal projection method to visualize 4D space.
result Direct synthetic construction of 3-sphere fibers from 2-sphere points.
Study shows moduli space of fibrations has specific homotopy types.
problem Understanding the homotopy types of moduli spaces of fibrations.
method Analyzes diffeomorphism group quotient and uses vector field models.
result Moduli space homotopy types are two-spheres or real projective planes.
Sub-Riemannian structures on odd-dimensional spheres respecting the Hopf fibration naturally appear in quantum mechanics. We study the curvature maps for such a sub-Riemannian structure and express them using the Riemannian curvature tensor of the Fubini-Study metric of the complex projective space and the curvature fo…
Study G2-metrics from non-integrable special Lagrangian fibrations.
problem Understanding G2-metrics from non-integrable special Lagrangian fibrations. method Decompose SU(3)-structures into solder 1-forms, connection 1-forms, and equivariant matrix-valued functions. result Describe regular parts of G2-manifolds with Lagrangian-type actions. We construct an explicit diffeomorphism taking any fibration of a sphere by great circles into the Hopf fibration, using elementary geometry--indeed the diffeomorphism is a local (differential) invariant, algebraic in derivatives.
Flat tori in 3-sphere have π extrinsic diameter under specific conditions.
problem Determining the extrinsic diameter of immersed flat tori in the 3-sphere.
method Analyzing asymptotic curves and Hopf fibration projections.
result The extrinsic diameter is π under certain topological conditions.
Harmonic and minimal great circle fibrations have special Gauss maps.
problem Characterizing Gauss maps of harmonic and minimal great circle fibrations.
method Analyzing the relationship between the Gauss map and the generating unit vector field.
result The Gauss map of a great circle fibration is harmonic (minimal) if and only if the generating unit vector field is harmonic (minimal).
In a 1983 paper with Frank Warner, we proved that the space of all great circle fibrations of the 3-sphere S^3 deformation retracts to the subspace of Hopf fibrations, and so has the homotopy type of a pair of disjoint two-spheres. Since that time, no generalization of this result to higher dimensions has been found, a…
Classifies foliations of complex and quaternionic projective spaces.
problem Classifying isoparametric foliations of complex and quaternionic projective spaces.
method Investigating projections of inhomogeneous isoparametric foliations of the 31-sphere under Hopf fibrations.
result Solved the last remaining open cases in the classification.
The study calculates the index distribution of Brownian loops in various geometrical settings.
problem Calculating the distribution of the index of Brownian loops in specific geometrical settings.
method Analysis based on the geometry of Hopf and anti-de Sitter fibrations, and the relationship between winding and area forms.
result Explicit formulas and asymptotics for the distribution of the index of the Brownian loop.
The paper extends Gluck and Warner's result on fibrations of spheres by great subspheres.
problem Understanding when two Hopf fibrations of S2n−1 agree on a fiber. method Characterizing the conditions for two Hopf fibrations of S2n−1 to agree on a fiber. result A complete characterization of the conditions for two Hopf fibrations of S2n−1 to agree on a fiber. New computations show sl(N) homology is related to SU(N) representations of knots.
problem Computing colored sl(N) homology for nontrivial knots and links.
method Using SU(N) representations of knot complements, we compute homology and show isomorphisms.
result Colored sl(N) homology is isomorphic to the cohomology of SU(N) representations of knot complements.
We study the horizontal Laplacian ΔH associated to the Hopf fibration S3→S2 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 ΔH. We express the Green functions for associated Poisson semigroup…
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…
We construct a geometric model of eight-dimensional manifolds and realize them in the context of type II string theory. These eight-manifolds are constructed by non-trivial T4 fibrations over Calabi-Yau two-folds. These give rise to eight-dimensional non-Kahler Hermitian manifolds with SU(4) structure. The eight…
We describe a family of locally conformal Kaehler metrics on class 1 Hopf surfaces H containing some recent metrics constructed by P. Gauduchon and L. ornea. We study some canonical foliations associated to these metrics, in particular a 2-dimensional foliation E that is shown to be independent of the metric. We elemen…
Complete integrability proved for SR geodesic flow on S^7.
problem Complete integrability of subriemannian geodesic flow on S^7.
method Adapting a method by A. Thimm, constructing functionally independent first integrals.
result Complete integrability in the sense of Liouville proved for SR geodesic flow.
Helix surfaces in Anti-de Sitter space maintain constant Gaussian curvature.
problem Understanding helix surfaces in Anti-de Sitter space with Berger-like metrics.
method Proved helix surfaces have constant Gaussian curvature and described them explicitly.
result Explicit local description of helix surfaces in terms of isometries and curves.
Paper presents a nearly invertible mapping between high-dimensional and lower-dimensional spheres.
problem Lack of rigorous mathematical foundation in deep learning algorithms.
method Utilizes Multicomplex rotation groups and polyspherical coordinates to define two maps and a composite LG Fibration.
result Derives a distance difference function to determine invariant inner products under the transformation.
In this survey, we remind some fibrations structure theorems (also called Milnor's fibrations) recently proved in the real and complex case, in the local and global settings. We give several Poincaré-Hopf type formulae which relates the Euler-Poincaré characteristic of these fibers (also called Milnor's fibers) and ind…
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.
Researchers classify geodesic orbit spaces for compact Lie groups of rank two.
problem Identifying geodesic orbit spaces for compact Lie groups of specific rank.
method Classification of simply connected geodesic orbit spaces where G is a compact Lie group of rank two.
result Only certain spheres and projective spaces, with metrics induced from Hopf fibrations, are geodesic orbit spaces for compact Lie groups of rank two.