The Poincare-hopf Theorem
نویسندگان
چکیده
Mapping degree, intersection number, and the index of a zero of a vector field are defined. The Poincare-Hopf theorem, which states that under reasonable conditions the sum of the indices of a vector field equals the Euler characteristic of the manifold, is proven. Some consequences are discussed.
منابع مشابه
A Quantum Analog of the Poincare–birkhoff–witt Theorem
We reduce the basis construction problem for Hopf algebras generated by skew-primitive semi-invariants to a study of special elements, called “super-letters,” which are defined by Shirshov standard words. In this way we show that above Hopf algebras always have sets of PBW-generators (“hard” super-letters). It is shown also that these Hopf algebras having not more than finitely many “hard” supe...
متن کاملA graph theoretical Poincare-Hopf Theorem
We introduce the index i(v) = 1− χ(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 ∈ S(v) | f(w) − f(v) < 0 } is the subgraph of the unit sphere S(v) generated by the vertices {w ∈ V | (v, w) ∈ E }. We prove that the sum ∑ v∈V i(v) of the indices always is equal to the Euler characteristic χ(G) of the graph G. This is...
متن کاملUniqueness of Generalized Equilibrium for Box Constrained Problems and Applications
In recent work, Simsek-Ozdaglar-Acemoglu [5] prove a generalization of the Poincare-Hopf Theorem and establish sufficient conditions for the uniqueness of generalized critical points (generalized equilibria) under some regularity assumptions. In this paper, we restrict ourselves to functions on box-constrained regions and establish the uniqueness of the generalized critical point with weaker re...
متن کاملStatic Equilibria of Rigid Bodies: Dice, Pebbles, and the Poincare-Hopf Theorem
By appealing to the Poincaré-Hopf Theorem on topological invariants, we introduce a global classification scheme for homogeneous, convex bodies based on the number and type of their equilibria. We show that beyond trivially empty classes all other classes are non-empty in the case of three-dimensional bodies; in particular we prove the existence of a body with just one stable and one unstable e...
متن کاملNormal forms of Hopf Singularities: Focus Values Along with some Applications in Physics
This paper aims to introduce the original ideas of normal form theory and bifurcation analysis and control of small amplitude limit cycles in a non-technical terms so that it would be comprehensible to wide ranges of Persian speaking engineers and physicists. The history of normal form goes back to more than one hundreds ago, that is to the original ideas coming from Henry Poincare. This tool p...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008