论文标题
$ f $ -neighbors Hopf型定理
Hopf-type Theorems For $f$-neighbors
论文作者
论文摘要
我们在一个程序的框架内工作，该程序旨在探索包含Borsuk-Ulam型定理类，一些固定点定理，KKM Lemma，Radon，Tverberg和Helly定理的类型的定理的各种扩展版本。在本文中，我们研究了HOPF定理的变体，这些变体是关于紧凑的Riemannian歧管$ n $ n $ to $ n $ to $ \ MATHBB {r}^n $的连续地图。我们使用$ n <m $调查了地图$ f \ colon m \ to \ mathbb {r}^m $，并介绍了几种不同类型的$ f $ -neighbors的概念，这是$ m $中的一对不同的点，因此$ f $将其带入了某种类型的“小”。每种类型的接下来，我们询问$ m $上的距离是什么距离，是这种类型的$ f $ -Neighbors之间的距离，并研究了这套距离的各种特征。我们的主要结果之一是如下。令$ f \ colon m \ to \ mathbb {r}^{m} $为连续地图。 We say that two distinct points $a$ and $b$ in $M$ are visual $f$-neighbors if the segment in $\mathbb{R}^{m}$ with endpoints $f(a)$ and $f(b)$ intersects $f(M)$ only at $f(a)$ and $f(b)$.然后，作为视觉$ f $ -Neighbors之间的距离实现的距离是无限的。此外，我们从定量的意义上概括了Hopf定理。
We work within the framework of a program aimed at exploring various extended versions for theorems from a class containing Borsuk-Ulam type theorems, some fixed point theorems, the KKM lemma, Radon, Tverberg, and Helly theorems. In this paper we study variations of the Hopf theorem concerning continuous maps of a compact Riemannian manifold $M$ of dimension $n$ to $\mathbb{R}^n$. We investigate the case of maps $f\colon M \to \mathbb{R}^m$ with $n < m$ and introduce several notions of varied types of $f$-neighbors, which is a pair of distinct points in $M$ such that $f$ takes it to a 'small' set of some type. Next for each type, we ask what distances on $M$ are realized as distances between $f$-neighbors of this type and study various characteristics of this set of distances. One of our main results is as follows. Let $f\colon M \to \mathbb{R}^{m}$ be a continuous map. We say that two distinct points $a$ and $b$ in $M$ are visual $f$-neighbors if the segment in $\mathbb{R}^{m}$ with endpoints $f(a)$ and $f(b)$ intersects $f(M)$ only at $f(a)$ and $f(b)$. Then the set of distances that are realized as distances between visual $f$-neighbors is infinite. Besides we generalize the Hopf theorem in a quantitative sense.
