Abstract

Geodesic nets on Riemannian manifolds form a natural class of stationary objects generalizing geodesics. Yet almost nothing is known about their classification or general properties even when the ambient Riemannian manifold is the Euclidean plane or the round 2-sphere. In the first part of this paper, we survey some results and open questions (old and new) about geodesic nets on Riemannian manifolds. Many of these open questions are about geodesic nets with edges of multiplicity one on the Euclidean plane. Our main focus is on relationships between the number of boundary vertices, the number of inner (or balanced) vertices, and some basic geometric characteristic of geodesic nets (such as the length or the imbalances at boundary vertices). The second part contains a new construction providing a partial answer for one of these questions: We describe an infinite family of geodesic nets with edges of multiplicity one on the Euclidean plane with a constant number (namely, 14) of boundary vertices and arbitrarily many inner (or balanced) vertices of degree $\ge 3.$ The fact that all edges of the constructed geodesic nets have multiplicity one is not proven but strongly supported by numerical evidence obtained from experimentation.

摘要

黎曼流形上的测地线网形成一类自然的静止对象来概括测地线。然而，即使当周围的黎曼流形是欧几里得平面或圆 2-球面时，关于它们的分类或一般性质几乎一无所知。在本文的第一部分，我们调查了一些关于黎曼流形上的测地线网的结果和未解决的问题（旧的和新的）。这些开放性问题中有许多是关于在欧几里得平面上具有多重性边缘的测地线网。我们主要关注边界顶点数、内部（或平衡）顶点数和测地线网的一些基本几何特征（例如边界顶点的长度或不平衡）之间的关系。第二部分包含一个新结构，为以下问题之一提供了部分答案：$\ge \mathrm{3个}.$构造的测地线网的所有边都具有多重性这一事实尚未得到证实，但得到了从实验中获得的数值证据的有力支持。