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 -sphere bundles.
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48 results for “Hopf degree theorem”
A proof based on the Chern-Gauss-Bonnet Theorem is given to Hopf Theorem concerning the degree of the Gauss map of a hypersurface in .
A new simple proof for surface map degree inequality.
problem Degree of maps between closed surfaces.
method Elementary proof without additional techniques.
result A new proof of the inequality χ(M) ≤ d·χ(N).
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…
Let G be a finite group. For semi-free G-manifolds which are oriented in the sense of Waner, the homotopy classes of G-equivariant maps into a G-sphere are described in terms of their degrees, and the degrees occurring are characterized in terms of congruences. This is first shown to be a stable problem and then solved…
Quantum invariants for fibered links determined by genus and Hopf invariant.
problem Quantum invariants of fibered links in .
method Genus bounds and Giroux-Goodman theorem on fiber surfaces.
result Top coefficient of ADO invariant is determined by Hopf invariant.
The paper sets genus bounds for twisted quantum invariants.
problem Bounding the degree of twisted quantum invariants for knots.
method Using Reshetikhin-Turaev construction and Drinfeld doubles.
result Degree of polynomials is bounded by .
Finite-type surfaces have a topological Hopf property.
problem Characterizing surfaces with a topological Hopf property.
method Using topological analogs of the Hopf property.
result Infinite-type surfaces do not have the Hopf property.
Global inverse function theorem proved easily using Riemannian geometry.
problem Global inverse function theorem in Riemannian geometry.
method Hopf--Rinow theorem in Riemannian geometry.
result Hadamard's global inverse function theorem is proven easily.
In this paper we compute the Leray Schauder degree for a fourth order elliptic boundary value problem with exponential nonlinearity and Navier boundary condition. This will be made by proving a Poincare'-Hopf type theorem. Moreover by using this result, together with some quantitative results about the formal set of ba…
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.
Proves positive Euler characteristic for certain manifolds with positive second intermediate Ricci curvature.
problem Hopf's conjecture on positive sectional curvature and its failure under relaxed conditions.
method Non-trivial extension of the Four Periodicity Theorem to higher degrees.
result Proves positive Euler characteristic for specific manifolds with positive second intermediate Ricci curvature.
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.
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 . In 1984 Jänich presented a Poincaré-Hopf th…
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 -spacetimes.
result Generalizes Hopf-Rinow theorem to a new class of semi-Riemannian manifolds.
Generalizes Poincaré-Hopf Theorem for piecewise smooth boundaries.
problem Conservation law for vector fields on surfaces with piecewise smooth boundaries.
method Generalization of the Poincaré-Hopf Theorem for real-analytic vector fields on surfaces with piecewise smooth boundaries.
result Conservation law for vector fields on surfaces with piecewise smooth boundaries.
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.
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.
In this paper we give an extension of the Cartier-Gabriel-Kostant structure theorem to Hopf algebroids.
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.
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.
This paper deals with a semi-classical limit (Theorem 1) by using traditional mathematical methods, and shows a Hopf theorem as a corollary. A formal discussion of it may be found in [7].
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.
The paper extends Hopf's theorem to convex surfaces and discrete triangulations.
problem Extending Hopf's theorem to convex surfaces and discrete triangulations.
method Investigates continuous maps and simplicial maps on convex polyhedra, proving theorems about neighbors and distances.
result The Hopf theorem and its quantitative generalization hold for convex surfaces, with quasigeodesics replacing geodesics.
Solves an old problem by showing round spheres are the only compact surfaces with specific curvature properties.
problem Finding compact surfaces in Euclidean 3-space with specific curvature properties.
method Representation of solutions to linear elliptic equations with discontinuous coefficients.
result Compact surfaces of genus zero with specific curvature properties are round spheres.
We show that simple coverings of B^4 branched over ribbon surfaces up to certain local ribbon moves bijectively represent orientable 4-dimensional 2-handlebodies up to handle sliding and addition/deletion of cancelling handles. As a consequence, we obtain an equivalence theorem for simple coverings of S^3 branched over…
Paper uses algebraic methods to prove an equivariant Poincaré-Hopf theorem.
problem Equivariant index theory on manifolds.
method Localization algebras and Witten deformation techniques in K-homology.
result Established an equivariant version of the Poincaré-Hopf theorem.
Extends E. Hopf's theorem to magnetic systems without conjugate points.
problem Proving magnetic curvature non-positive for magnetic systems without conjugate points.
method Using magnetic curvature introduced by the first author, proving magnetic flatness conditions.
result Magnetic flatness is a rigid condition with specific metric and curvature properties.
Symmetric hypersurfaces with constant mean curvature are spheres.
problem Symmetry of hypersurfaces with constant mean curvature.
method Exposition of results on symmetry properties and variations of the Hopf Lemma.
result Symmetric hypersurfaces with constant mean curvature are spheres.
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.
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.
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.
We treat the Witten operator on the de Rham complex with semiclassical heat kernel methods to derive the Poincaré-Hopf theorem and degenerate generalizations of it. Thereby, we see how the semiclassical asymptotics of the Witten heat kernel are related to approaches using the Thom form of Mathai and Quillen.
An locally conformally Kahler (LCK) manifold with potential is a complex manifold with a cover which admits an automorphic Kahler potential. An LCK manifold with potential can be embedded to a Hopf manifold, if its dimension is at least 3. We give a functional-analytic proof of this result based on Riesz-Schauder theor…
Let M be a closed 3-manifold which can be triangulated with N simplices. We prove that any map from M to a genus 2 surface has Hopf invariant at most C^N. Let X be a closed oriented hyperbolic 3-manifold with injectivity radius less than epsilon at one point. If there is a degree non-zero map from M to X, then we prove…
Extends Adams' theorem to periodic cohomology.
problem Proving Adams' theorem for periodic cohomology.
method Adapting Adams' approach to periodic cohomology.
result Conjecture proven in a special case.
We formulate and prove an analog of the Hopf Index Theorem for Riemannian foliations. We compute the basic Euler characteristic of a closed Riemannian manifold as a sum of indices of a non-degenerate basic vector field at critical leaf closures. The primary tool used to establish this result is an adaptation to foliati…
We prove a uniqueness theorem for immersed spheres of prescribed (non-constant) mean curvature in homogeneous three-manifolds. In particular, this uniqueness theorem proves a conjecture by A.D. Alexandrov about immersed spheres of prescribed Weingarten curvature in R3 for the special but important case of prescribed me…
For a manifold with boundary, the restriction of Chern's transgression form of the Euler curvature form over the boundary is closed. Its cohomology class is called the secondary Chern-Euler class and used by Sha to formulate a relative Poincaré-Hopf theorem, under the condition that the metric on the manifold is locall…
The goal of this work is to generalize the Gauss-Bonnet and Poincaré-Hopf Theorems to the case of orbifolds with boundary. We present two such generalizations, the first in the spirit of Satake. In this case, the local data (i.e. integral of the curvature in the case of the Gauss-Bonnet Theorem and the index of the vec…
In this note, we consider generalizations of the asymptotic Hopf invariant, or helicity, for Hamiltonian systems with one-and-a-half degrees of freedom and symplectic diffeomorphisms of a two-disk to itself.
The quadratic linking degreemath.AG
Introduces quadratic linking degree in algebraic geometry.
problem No specific problem stated; focuses on new concept.
method Motivic homotopy theory and Witt group.
result Explicit computation of quadratic linking degree.
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.
Hopf type theorems for surfaces in the de Sitter-Schwarzschild and Reissner-Nordstrom manifoldsmath.DG
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.
Study magnetic geodesics on half-Lie groups, proving Hopf-Rinow theorem for energies above critical value.
problem Investigate magnetic geodesics on half-Lie groups using Riemannian and two-form structures.
method Define Mañé's critical value, prove Finsler geodesic flow equivalence, and apply Hopf-Rinow theorem.
result Hopf-Rinow theorem holds for energies above Mañé's critical value on magnetic geodesics.
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.
Let be an oriented link with an alternating diagram . It is known that is a fibered link if and only if the surface obtained by applying Seifert's algorithm to is a Hopf plumbing. Here, we call a Hopf plumbing if is obtained by successively plumbing finite number of Hopf bands to a disk. In t…
We introduce a method, based on the Poincare-Hopf index theorem, to classify solutions to overdetermined problems for fully nonlinear elliptic equations in domains diffeomorphic to a closed disk. Applications to some well-known nonlinear elliptic PDEs are provided. Our result can be seen as the analogue of Hopf's uniqu…