arXiv research

Generalizes Hopf degree theorem to nontrivial bundles.

Global inverse function theorem proved easily using Riemannian geometry.

The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.

The Poincaré-Hopf theorem is extended to 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 $2k \geq 4$. In 1984 Jänich presented a Poincaré-Hopf th…

Extends Hopf-Rinow theorem to semi-Riemannian spacetimes.

Generalizes Poincaré-Hopf Theorem for piecewise smooth boundaries.

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

Hopf's theorem generalized to curved spaces.

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 $\mathbb{R}^n$.

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.

Holomorphic vector bundles on Hopf manifolds admit flat connections.

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.

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

Paper uses algebraic methods to prove an equivariant Poincaré-Hopf theorem.

Extends E. Hopf's theorem to magnetic systems without conjugate points.

Symmetric hypersurfaces with constant mean curvature are spheres.

New proofs for complex Hopf manifolds using geometric structures.

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

Extends Tian theorem to Vaisman manifolds for approximations.

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…

Extends Adams' theorem to periodic cohomology.

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…

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

Extends Hopf's theorem to de Sitter-Schwarzschild and Reissner-Nordstrom manifolds.

Study magnetic geodesics on half-Lie groups, proving Hopf-Rinow theorem for energies above critical value.

Supports conjecture about harmonic maps from S³ to S².

Let $L$ be an oriented link with an alternating diagram $D$. It is known that $L$ is a fibered link if and only if the surface $R$ obtained by applying Seifert's algorithm to $D$ is a Hopf plumbing. Here, we call $R$ a Hopf plumbing if $R$ 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…

The paper studies coherent sheaves on subvarieties of Hopf manifolds.

Study of $f$-neighbors in Riemannian manifolds, proving infinite set of distances.

Geodesic connectedness proved for statistical manifolds with divisible cubic forms.

Link groups can only have certain SU(2) representations.

Paper proves non-existence of certain hypersurfaces in complex quadric.

In this paper a geometric approach toward stable homotopy groups of spheres, based on the Pontrjagin-Thom construction is proposed. From this approach a new proof of Hopf Invariant One Theorem by J.F.Adams for all dimensions except $15,31,63,127$ is obtained. It is proved that for $n>127$ in the stable homotopy group o…

Proves uniqueness of capillary disks in 3D domains.

The paper classifies real hypersurfaces with a specific Jacobi operator in complex Grassmannians.

In [1], Theorem 3, the authors proved, in one dimension, a generalization of the Hopf Lemma, and the question arose if it could be extended to higher dimensions. In this paper we present two conjectures as possible extensions, and give a very partial answer. We write this paper to call attention to the problem.

The paper studies bifurcations in discrete dynamical systems on manifolds.

Study vector fields on non-compact manifolds with group action.

This paper is a continuation of the paper F. A. Arias and M. Malakhaltsev "A generalization of the Gauss-Bonnet and Hopf-Poincaré theorems", ArXiv:1510.01395 [MathDG] 5 Oct 2015. Let $π: E \to M$ be a locally trivial fiber bundle over a two-dimensional manifold $M$, and $Σ\subset M$ be a discrete subset. A subset $Q \s…