A generalized Poincaré-Hopf index theorem

The Poincaré-Hopf Theorem for line fields revisited

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 theorem for line fields with more complicated singularities and focussed on the complexities arising in the generalised setting. In this expository note we review the Poincaré-Hopf Theorem for line fields with point singularities, presenting a careful proof which is valid in all di…

The theorems of Green-Stokes,Gauss-Bonnet and Poincare-Hopf in Graph Theory

By proving graph theoretical versions of Green-Stokes, Gauss-Bonnet and Poincare-Hopf, core ideas of undergraduate mathematics can be illustrated in a simple graph theoretical setting. In this pedagogical exposition we present the main proofs on a single page and add illustrations. While discrete Stokes is is old, the other two results for graphs were found only recently.

A generalization of the Gauss-Bonnet and Hopf-Poincaré theorems

We consider a locally trivial fiber bundle $π: E \to M$ over a compact oriented two-dimensional manifold $M$, and a section $s$ of this bundle defined over $M \setminus Σ$, where $Σ$ is a discrete subset of $M$. We call the set $Σ$ the set of singularities of the section $s : M \setminus Σ\to E$. We assume that the behavior of the section $s$ at the singularities is controlled in the following way: $s(M \setminus Σ)$ coincides with the interior part of a surface $S \subset E$ with boundary $\partial S$, and $\partial S$ is $π^{-1}(Σ)$. For such sections $s$ we define an index of $s$ at a point…

A generalisation of the Poincaré-Hopf Theorem

The Poincaré-Hopf Theorem is a conservation law for real-analytic vector fields, which are tangential to a closed surface (such as a torus or a sphere). The theorem also governs real-analytic vector fields, which are tangential to surfaces with smooth boundaries; in these cases, the vector field must be pointing in the outward normal direction along the boundary. In this paper, I will generalise the Poincaré-Hopf Theorem for real-analytic vector fields that are tangential to surfaces with piecewise smooth boundaries, and not parallel to the outward normal of the boundary.

指标与留数：从庞加莱-霍普夫（Poincaré-Hopf）到鲍姆-博特（Baum-Bott），以及马尔科·布鲁内拉（Marco Brunella）

Indices and residues: from Poincaré-Hopf to Baum-Bott, and Marco Brunella

该综述性文章研究向量场与全纯叶状结构的不变量，旨在关联复解析奇异簇与复流形上奇异全纯叶状结构这两个有诸多共通点的理论领域。

共形Hopf映射的指标

The index of conformal Hopf maps

该研究计算从三维球面到二维球面的任意调和态射的指标与零性

A graph theoretical Poincare-Hopf Theorem

We introduce the index i(v) = 1 - X(S(v)) for critical points of a locally injective function f on the vertex set V of a simple graph G=(V,E). Here S(v) = {w in E | (v,w) in E, f(w)-f(v)<0} is the subgraph of the unit sphere at v in G. It is the exit set of the gradient vector field. We prove that the sum of i(v) over V is always is equal to the Euler characteristic X(G) of the graph G. This is a discrete Poincare-Hopf theorem in a discrete Morse setting. It allows to compute X(G) for large graphs for which other methods become impractical.

Hopf algebras, cyclic cohomology and the transverse index theory

We present the solution of a longstanding internal problem of noncommutative geometry, namely the computation of the index of a transversally elliptic operator on an arbitrary foliation. The new and crucial ingredient is a certain Hopf algebra associated to the transverse frame bundle. Its cyclic cohomology is defined and shown to be canonically isomorphic to the Gelfand-Fuks cohomology.

On Sha's secondary Chern-Euler class

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 locally product near the boundary. We show that the secondary Chern-Euler form is exact away from the outward and inward unit normal vectors of the boundary by explicitly constructing a transgression form. Using Stokes' theorem, this evaluates the boundary term in Sha's relative Poinca…

等变 Poincaré-Hopf 定理

Equivariant Poincaré-Hopf theorem

本文利用局部化代数计算 Euler 特征算子的等变 K-同调类，结合 Witten 形变技术建立了 Poincaré-Hopf 定理的等变版本，推广了 Lück-Rosenberg 的经典结果。

Poincaré index and the volume functional of unit vector fields on punctured spheres

For $n\geq 1$, we exhibit a lower bound for the volume of a unit vector field on $\mathbb{S}^{2n+1}\backslash\{\pm p\}$ depending on the absolute values of its Poincaré indices around $\pm p$. We determine which vector fields achieve this volume, and discuss the idea of having multiple isolated singularities of arbitrary configurations.

二维紧流形上 Filippov 向量场的 Poincaré-Hopf 定理

Poincaré-Hopf Theorem for Filippov vector fields on 2-dimensional compact manifolds

本文提出涵盖伪平衡点与切向奇点的新指数定义，证明带光滑切换流形的二维紧流形上 Filippov 向量场满足 Poincaré-Hopf 定理，并推导 Hairy Ball 定理变体。

On the biharmonic and harmonic indices of the Hopf map

Biharmonic maps are the critical points of the bienergy functional and, from this point of view, generalise harmonic maps. We consider the Hopf map $ψ:\s^3\to \s^2$ and modify it into a nonharmonic biharmonic map $ϕ:\s^3\to \s^3$. We show $ϕ$ to be unstable and estimate its biharmonic index and nullity. Resolving the spectrum of the vertical Laplacian associated to the Hopf map, we recover Urakawa's determination of its harmonic index and nullity.

Mall bundles and flat connections on Hopf manifolds

A Mall bundle on a Hopf manifold H is a holomorphic vector bundle whose pullback to the universal cover of H is trivial. We define resonant and non-resonant Mall bundles, generalizing the notion of the resonance in ODE, and prove that a non-resonant Mall bundle always admits a flat holomorphic connection. We use this observation to prove a version of Poincare-Dulac linearization theorem, showing that any non-resonant invertible holomorphic contraction of a complex space is linear in appropriate holomorphic coordinates. We define the notion of resonance in Hopf manifolds, and show that all non-…

A Hopf Index Theorem for foliations

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 foliations of the Witten deformation method.

Hopf 流形上的全纯几何结构

Holomorphic geometric structures on Hopf manifolds

本文在不依赖 Poincaré-Dulac 正规形式的情况下，于所有复 Hopf 流形上构造了可积全纯 G-结构与平坦全纯 Cartan 几何，并借助适配坐标图给出了该定理的新证明。

基于庞加莱-霍普夫定理与流形理论的网络双病毒流行病模型平衡点分析

Equilibria analysis of a networked bivirus epidemic model using Poincaré--Hopf and Manifold Theory

本文针对网络双病毒SIS模型共存平衡点结果有限的问题，通过改进庞加莱-霍普夫定理并结合Morse不等式，给出了共存平衡点数量的下界及其局部稳定性性质，并用数值例子验证了多个吸引子与共存平衡点的存在。

关于与 Hopf 和 anti-de Sitter 纤维化相关的布朗环路指数的定律

On the law of the index of Brownian loops related to the Hopf and anti-de Sitter fibrations

本文基于 Hopf 和 anti-de Sitter 纤维化的几何及环绕数与面积形式的关系，给出了复射影直线、复双曲直线、奇数维球面和复 anti-de Sitter 空间等几何设定下布朗环路指数分布的显式公式与渐近性质。

Overdetermined elliptic problems in topological disks

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 uniqueness theorem for constant mean curvature spheres, but for the general analytic context of overdetermined elliptic problems.

Geometric construction of Hopf cyclic characteristic classes

In earlier joint work with A. Connes on transverse index theory on foliations, cyclic cohomology adapted to Hopf algebras has emerged as a decisive tool in deciphering the total index class of the hypoelliptic signature operator. We have found a Hopf algebra H(n), playing the role of a `quantum structure group' for the `space of leaves' of a codimension n foliation, whose Hopf cyclic cohomology is canonically isomorphic to the Gelfand-Fuks cohomology of the Lie algebra of formal vector fields. However, with a few low-dimensional exceptions, no explicit construction was known for its Hopf cycli…