A Poincaré-Hopf Type Formula for A Pair of Vector Fields

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…

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。

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。

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。

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。

$E(\kappa,\tau)$ 中 Cauchy-Riemann 不等式下的球面与非常均曲率 Hopf 环面

Spheres under a Cauchy-Riemann inequality in $E(κ,τ)$, and non-CMC Hopf tori

本文证明在齐性空间 $E(\kappa,\tau)$ 中，球面浸入满足单侧 Cauchy-Riemann 型不等式即可推出平均曲率为常数，并构造了 Berger 球面上的非-CMC Hopf 环面反例，同时给出闭曲面为环面的判别条件。

A d'Alembert Formula for Hopf Hypersurfaces

A Hopf hypersurface in complex hyperbolic space CH^n is one in which the complex structure applied to the normal vector is a principal direction at each point. In this paper, Hopf hypersurfaces for which the corresponding principal curvature is small (relative to the ambient sectional curvature) are studied by means of a generalized Gauss map into a product of spheres, and it is shown that the hypersurface may be recovered from the image of this map, via an explicit formula.